THREE DIMENSIONAL GEOMETRY

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Chapter 11 THREE DIMENSIONAL GEOMETRY 111 Overview 1111 Direction cosines of a line are the cosines of the angles made by the line with positive directions of the co-ordinate axes 111 If l, m, n are the direction cosines of a line, then l + m + n = Direction cosines of a line joining two points P (x 1, y 1, z 1 ) and Q (x, y, z ) are x x 1 1 1, y y, z z, PQ PQ PQ where PQ= ( x x ) +( y y ) + ( z z ) Direction ratios of a line are the numbers which are proportional to the direction cosines of the line 1115 If l, m, n are the direction cosines and a, b, c are the direction ratios of a line, then l ; m ; n 1116 Skew lines are lines in the space which are neither parallel nor interesecting They lie in the different planes 1117 Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines 1118 If l 1, m 1, n 1 and l, m, n are the direction cosines of two lines and θ is the acute angle between the two lines, then cosθ = ll 1 + mm 1 + nn If a 1, b 1, c 1 and a, b, c are the directions ratios of two lines and θ is the acute angle between the two lines, then

2 THREE DIMENSIONAL GEOMETRY 1 cosθ= aa + bb + cc a a a b b b Vector equation of a line that passes through the given point whose position vector is a and parallel to a given vector b is r= a+λb Equation of a line through a point (x 1, y 1, z 1 ) and having directions cosines l, m, n (or, direction ratios a, b and c) is x x1 y y1 z z1 = = or l m n x x1 y y1 z z1 = = 1111 The vector equation of a line that passes through two points whose positions vectors are a and b is r= a+λ( b a) Cartesian equation of a line that passes through two points (x 1, y 1, z 1 ) and (x, y, z ) is x x1 y y1 z z1 = = x x y y z z If θ is the acute angle between the lines r = a1+λb 1 b1 b b 1 1 b θ is given by cosθ= or θ= cos b b b b 1 1 and r = a+λb, then x x1 y y1 z z1 x x y y z z If = = l1 m1 n and = = 1 l1 m n are equations of two lines, then the acute angle θ between the two lines is given by cosθ = ll 1 + mm 1 + nn The shortest distance between two skew lines is the length of the line segment perpendicular to both the lines The shortest distance between the lines r = a1+λb and 1 r = a+λb is

3 MATHEMATICS b1 b a a1 b b Shortest distance between the lines: x x1 y y1 z z1 = = and x x y y z z = = is x x y y z z ( bc bc) ( ca c a) ( ab a b) Distance between parallel lines r a1 b and r = a +λb is b a a1 b 1110 The vector equation of a plane which is at a distance p from the origin, where ˆn is the unit vector normal to the plane, is r nˆ = p 1111 Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n is lx + my + nz = p 111 The equation of a plane through a point whose position vector is a and perpendicular to the vector n is ( r a) n= 0or rn = d, where d = a n 1113 Equation of a plane perpendicular to a given line with direction ratios a, b, c and passing through a given point (x 1, y 1, z 1 ) is a (x x 1 ) + b (y y 1 ) + c (z z 1 ) = Equation of a plane passing through three non-collinear points (x 1, y 1, z 1 ), (x, y, z ) and (x 3, y 3, z 3 ) is

4 THREE DIMENSIONAL GEOMETRY 3 x x1 y y1 z z1 x x1 y y1 z z1 = 0 x x y y z z Vector equation of a plane that contains three non-collinear points having position vectors a, b, c is ( r a) ( b a) ( c a) = Equation of a plane that cuts the co-ordinates axes at (a, 0, 0), (0, b, 0) and x y z (0, 0, c ) is + + = Vector equation of any plane that passes through the intersection of planes r n1= d1and r n= d is ( r n1 d1) +λ( r n d) = 0, where λ is any non-zero constant 1118Cartesian equation of any plane that passes through the intersection of two given planes A 1 x + B 1 y + C 1 z + D 1 = 0 and A x + B y + C z + D = 0 is (A 1 x + B 1 y + C 1 z + D 1 ) + λ ( A x + B y + C z + D ) = Two lines r a1 b and 1 r = a+λb are coplanar if ( a a1)( b1 b) = 0 x x1 y y1 z z Two lines x x y y z z and are coplanar if x x y y z z = 0, 11131In vector form, if θ is the acute angle between the two planes, r n1 = d1 and 1 n1 n r n = d, then θ= cos n n The acute angle θ between the line r a b and plane r n = d is given by

5 4 MATHEMATICS sin θ= b n b n 11 Solved Examples Short Answer (SA) Example 1 If the direction ratios of a line are 1, 1,, find the direction cosines of the line Solution The direction cosines are given by l=, m =, n= a + b + c a + b + c a + b + c Here a, b, c are 1, 1,, respectively Therefore, l= 1, m = 1, n= ie, 1 1 l=, m=, n= ie ±,, are DC s of the line Example Find the direction cosines of the line passing through the points P (, 3, 5) and Q ( 1,, 4) Solution The direction cosines of a line passing through the points P (x 1, y 1, z 1 ) and Q (x, y, z ) are x x, y y, z z PQ PQ PQ Here PQ = ( x x ) + ( y y ) + ( z z ) = ( 1 ) + ( 3) + (4 5) = = 11 Hence DC s are

6 THREE DIMENSIONAL GEOMETRY 5 ± 3 1 1,, or ±,, Example 3 If a line makes an angle of 30, 60, 90 with the positive direction of x, y, z-axes, respectively, then find its direction cosines Solution The direction cosines of a line which makes an angle of α, β, γ with the axes, are cosα, cosβ, cosγ Therefore, DC s of the line are cos30, cos60, cos90 ie, 3 1 ±,, 0 Example 4 The x-coordinate of a point on the line joining the points Q (,, 1) and R (5, 1, ) is 4 Find its z-coordinate Solution Let the point P divide QR in the ratio λ : 1, then the co-ordinate of P are But x coordinate of P is 4 Therefore, 5λ+ = 4 λ= λ+ 1 5λ+ λ+ λ+ 1,, λ+ 1 λ+ 1 λ+ 1 Hence, the z-coordinate of P is λ+ 1 = 1 λ+ 1 Example 5 Find the distance of the point whose position vector is ( iˆ+ ˆj kˆ ) from the plane r ( î ĵ + 4 ˆk ) = 9 Solution Here a = iˆ+ ˆj kˆ, n iˆ ˆj 4kˆ and d = 9 ( iˆ ˆj kˆ ) ( iˆ ˆj kˆ ) So, the required distance is

7 6 MATHEMATICS = = 1 1 Example 6 Find the distance of the point (, 4, 5) from the line Solution Here P (, 4, 5) is the given point Any point Q on the line is given by (3λ 3, 5λ + 4, (6λ 8 ), x + 3 y 4 z + 8 = = Since PQ = (3λ 1) PQ ( 3iˆ 5ˆj 6kˆ ) + +, we have iˆ+ 5 λ ˆj+ (6λ 3) kˆ 3 (3λ 1) + 5( 5λ) λ) + 6 (6λ 3 ) = 0 9λ + 5λ + 36λ = 1, ie λ = 3 10 Thus Hence PQ = i ˆ + ˆ j k ˆ PQ = = Example 7 Find the coordinates of the point where the line through (3, 4, 5) and (, 3, 1) crosses the plane passing through three points (,, 1), (3, 0, 1) and (4, 1, 0) Solution Equation of plane through three points (,, 1), (3, 0, 1) and (4, 1, 0) is ie ( r (iˆ+ ˆj+ kˆ) ( iˆ ˆj) ( iˆ ˆj kˆ) = 0 r( iˆ+ ˆj+ kˆ ) = 7 or x + y + z 7 = 0 (1) Equation of line through (3, 4, 5) and (, 3, 1) is x 3 y + 4 z + 5 = = ()

8 THREE DIMENSIONAL GEOMETRY 7 Any point on line () is ( λ + 3, λ 4, 6λ 5) This point lies on plane (1) Therefore, ( λ + 3) + (λ 4) + (6λ 5) 7 = 0, ie, λ = z Hence the required point is (1,, 7) Long Answer (LA) Example 8 Find the distance of the point ( 1, 5, 10) from the point of intersection of the line r= iˆ ˆj+ kˆ+λ (3iˆ+ 4ˆj+ kˆ ) and the plane r( iˆ ˆj+ kˆ ) = 5 Solution We have r= iˆ ˆj+ kˆ+λ (3iˆ+ 4ˆj+ kˆ) and r( iˆ ˆj+ kˆ ) = 5 Solving these two equations, we get [(iˆ ˆj+ kˆ) +λ (3iˆ+ 4 ˆj+ kˆ)]( iˆ ˆj+ kˆ) = 5 which gives λ = 0 Therefore, the point of intersection of line and the plane is (, 1,) and the other given point is ( 1, 5, 10) Hence the distance between these two points is ( 1) [ 1 5] [ ( 10)], ie 13 Example 9 A plane meets the co-ordinates axis in A, B, C such that the centroid of the Δ ABC is the point (α, β, γ) Show that the equation of the plane is x y z + + = 3 α β γ Solution Let the equation of the plane be x a + y b + z c = 1 Then the co-ordinate of A, B, C are (a, 0, 0), (0,b,0) and (0, 0, c) respectively Centroid of the Δ ABC is x x x, y y y, z z z ie,, But co-ordinates of the centroid of the Δ ABC are (α, β, γ) (given)

9 8 MATHEMATICS Therefore, α = 3 a, β = 3 b, γ = 3 c, ie a = 3α, b = 3β, c = 3γ Thus, the equation of plane is x y z + + = 3 α β γ Example 10 Find the angle between the lines whose direction cosines are given by the equations: 3l + m + 5n = 0 and 6mn nl + 5lm = 0 Solution Eliminating m from the given two equations, we get n + 3 ln + l = 0 (n + l) (n + l) = 0 either n = l or l = n Now if l = n, then m = n and if l = n, then m = n Thus the direction ratios of two lines are proportional to n, n, n and n, n, n, ie 1,, 1 and, 1, 1 So, vectors parallel to these lines are a = i + j k and If θ is the angle between the lines, then cos θ = a b a b b = i + j + k, respectively = ( i+ j k) i+ j+ k ( ) ( 1) ( ) = 1 6 Hence θ = cos 1 1 6

10 THREE DIMENSIONAL GEOMETRY 9 Example 11 Find the co-ordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line joining the points B (0, 1, 3) and C (, 3, 1) Solution Let L be the foot of perpendicular drawn from the points A (1, 8, 4) to the line passing through B and C as shown in the Fig 11 The equation of line BC by using formula r = a + λ ( b a ), the equation of the line BC is r = ( j+ 3k ) +λ i j 4k ( ) xi yi zk = i 1 i 3 4 k Comparing both sides, we get x = λ, y = (λ + 1), z = 3 4λ (1) Thus, the co-ordinate of L are (λ, (λ + 1), (3 4λ), so that the direction ratios of the line AL are (1 λ), 8 + (λ + 1), 4 (3 4λ), ie 1 λ, λ + 9, 1 + 4λ Since AL is perpendicular to BC, we have, (1 λ) ( 0) + (λ + 9) ( 3 + 1) + (4λ + 1) ( 1 3) = 0 λ = 5 6 The required point is obtained by substituting the value of λ, in (1), which is

11 30 MATHEMATICS 5 19,, Example 1 Find the image of the point (1, 6, 3) in the line x y 1 z 1 3 Solution Let P (1, 6, 3) be the given point and let L be the foot of perpendicular from P to the given line The coordinates of a general point on the given line are x 0 y 1 z 1 3, ie, x = λ, y = λ + 1, z = 3λ + If the coordinates of L are (λ, λ + 1, 3λ + ), then the direction ratios of PL are λ 1, λ 5, 3λ 1 But the direction ratios of given line which is perpendicular to PL are 1,, 3 Therefore, (λ 1) 1 + (λ 5) + (3λ 1) 3 = 0, which gives λ = 1 Hence coordinates of L are (1, 3, 5) Let Q (x 1, y 1, z 1 ) be the image of P (1, 6, 3) in the given line Then L is the mid-point x1 1 y1 6 z1 3 of PQ Therefore, 1, 3 5 x 1 = 1, y 1 = 0, z 1 = 7 Hence, the image of (1, 6, 3) in the given line is (1, 0, 7)

12 THREE DIMENSIONAL GEOMETRY 31 Example 13 Find the image of the point having position vector i 3 j 4 k in the plane r i j k 3 0 Solution Let the given point be P i 3j 4k and Q be the image of P in the plane r i j k 3 0 as shown in the Fig 114 Then PQ is the normal to the plane Since PQ passes through P and is normal to the given plane, so the equation of PQ is given by r = i 3j 4k i j k Since Q lies on the line PQ, the position vector of Q can be expressed as λ i+ 3 λ j+ 4+ λ k i j k i j k, ie, ( ) ( ) ( ) Since R is the mid point of PQ, the position vector of R is ( 1+ λ ) i+ ( 3 λ ) j+ ( 4+λ ) k + i+ 3 j+ 4k

13 3 MATHEMATICS ie, λ ( 1) 3 λ λ+ i+ j 4 k + + Again, since R lies on the plane r ( i j+ k) + 3= 0, we have λ ( 1) 3 j λ 4 λ+ i+ + + k ( i j k + ) + 3= 0 λ = Hence, the position vector of Q is ( i + 3j + 4k) i j k, ie 3 i+ 5j + k Objective Type Questions Choose the correct answer from the given four options in each of the Examples 14 to 19 Example 14 The coordinates of the foot of the perpendicular drawn from the point (, 5, 7) on the x-axis are given by (A) (, 0, 0) (B) (0, 5, 0) (C) (0, 0, 7) (D) (0, 5, 7) Solution (A) is the correct answer Example 15 P is a point on the line segment joining the points (3,, 1) and (6,, ) If x co-ordinate of P is 5, then its y co-ordinate is (A) (B) 1 (C) 1 (D) Solution (A) is the correct answer Let P divides the line segment in the ratio of λ : 1, x - coordinate of the point P may be expressed as λ = Thus y-coordinate of P is λ+ = λ+ 1 6λ+ 3 x = giving 6 λ+ 3 = 5 λ+ 1 λ+ 1 so that Example 16 If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are (A) sin α, sin β, sin γ (B) cos α, cos β, cos γ (C) tan α, tan β, tan γ (D) cos α, cos β, cos γ

14 THREE DIMENSIONAL GEOMETRY 33 Solution (B) is the correct answer Example 17 The distance of a point P (a, b, c) from x-axis is (A) a c (B) a b (C) b c (D) b + c Solution (C) is the correct answer The required distance is the distance of P (a, b, c) from Q (a, o, o), which is b c Example 18 The equations of x-axis in space are (A) x = 0, y = 0 (B) x = 0, z = 0 (C) x = 0 (D) y = 0, z = 0 Solution (D) is the correct answer On x-axis the y- co-ordinate and z- co-ordinates are zero Example 19 A line makes equal angles with co-ordinate axis Direction cosines of this line are (A) ± (1, 1, 1) (B) ±,, (C) ±,, (D) ±,, Solution (B) is the correct answer Let the line makes angle α with each of the axis Then, its direction cosines are cos α, cos α, cos α Since cos α + cos α + cos α = 1 Therefore, cos α = 1 3 Fill in the blanks in each of the Examples from 0 to Example 0 If a line makes angles its direction cosines are 3, 4 and 4 with x, y, z axis, respectively, then

15 34 MATHEMATICS Solution The direction cosines are cos, cos 3 4, cos 4, ie, 1 1 ± 0, Example 1 If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin α + sin β + sin γ is Solution Note that sin α + sin β + sin γ = (1 cos α) + (1 cos β) + (1 cos γ) = 3 (cos α + cos β + cos γ) = Example If a line makes an angle of 4 with each of y and z axis, then the angle which it makes with x-axis is Solution Let it makes angle α with x-axis Then cos α + cos + 4 cos = 1 4 which after simplification gives α = State whether the following statements are True or False in Examples 3 and 4 Example 3 The points (1,, 3), (, 3, 4) and (7, 0, 1) are collinear Solution Let A, B, C be the points (1,, 3), (, 3, 4) and (7, 0, 1), respectively Then, the direction ratios of each of the lines AB and BC are proportional to 3, 1, 1 Therefore, the statement is true Example 4 The vector equation of the line passing through the points (3,5,4) and (5,8,11) is r 3iˆ 5ˆj 4kˆ ( iˆ 3ˆj 7kˆ) Solution The position vector of the points (3,5,4) and (5,8,11) are r r a 3iˆ 5ˆj 4kb ˆ, 5iˆ 8ˆj 11k ˆ, and therefore, the required equation of the line is given by r 3iˆ 5ˆj 4kˆ ( iˆ 3ˆj 7kˆ) Hence, the statement is true

16 THREE DIMENSIONAL GEOMETRY EXERCISE Short Answer (SA) 1 Find the position vector of a point A in space such that OA is inclined at 60º to OX and at 45 to OY and OA = 10 units Find the vector equation of the line which is parallel to the vector 3iˆ ˆj 6kˆ and which passes through the point (1,,3) 3 Show that the lines x 1 y z x 4 y 1 and = = z intersect 5 Also, find their point of intersection 4 Find the angle between the lines r = 3iˆ ˆj+ 6 kˆ+ λ(iˆ+ ˆj+ kˆ r ) and = ( ˆj 5 kˆ) + μ(6iˆ+ 3ˆj+ kˆ) 5 Prove that the line through A (0, 1, 1) and B (4, 5, 1) intersects the line through C (3, 9, 4) and D ( 4, 4, 4) 6 Prove that the lines x = py + q, z = ry + s and x = p y + q, z = r y + s are perpendicular if pp + rr + 1 = 0 7 Find the equation of a plane which bisects perpendicularly the line joining the points A (, 3, 4) and B (4, 5, 8) at right angles 8 Find the equation of a plane which is at a distance 3 3 units from origin and the normal to which is equally inclined to coordinate axis 9 If the line drawn from the point (, 1, 3) meets a plane at right angle at the point (1, 3, 3), find the equation of the plane 10 Find the equation of the plane through the points (, 1, 0), (3,, ) and (3, 1, 7)

17 36 MATHEMATICS 11 Find the equations of the two lines through the origin which intersect the line x 3 y 3 z π = = at angles of each Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l + m n = 0 13 If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ = δl + δm + δn 14 O is the origin and A is (a, b, c)find the direction cosines of the line OA and the equation of plane through A at right angle to OA 15 Two systems of rectangular axis have the same origin If a plane cuts them at distances a, b, c and a, b, c, respectively, from the origin, prove that = a b c Long Answer (LA) 16 Find the foot of perpendicular from the point (,3, 8) to the line 4 x y 1 z Also, find the perpendicular distance from the given point 6 3 to the line 17 Find the distance of a point (,4, 1) from the line x 5 y 3 z Find the length and the foot of perpendicular from the point 1,, to the plane x y + 4z + 5 = 0 19 Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + y = 0 and 3y z = 0

18 THREE DIMENSIONAL GEOMETRY 37 0 Find the equation of the plane through the points (,1, 1) and ( 1,3,4), and perpendicular to the plane x y + 4z = 10 1 Find the shortest distance between the lines given by r= (8+ 3 λiˆ (9+ 16 λ) ˆj+ ˆ ( λ ) kand r= 15iˆ+ 9 ˆj+ 5 kˆ+μ (3iˆ+ 8 ˆj 5 kˆ) Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + y + 3z 4 = 0 and x + y z + 5 = 0 3 The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α Prove that the equation of the plane in its new position is ax + by ± ( a + b tan α) z = 0 4 Find the equation of the plane through the intersection of the planes r ( î + 3 ĵ ) 6 = 0 and r (3 î ĵ 4 ˆk ) = 0, whose perpendicular distance from origin is unity 5 Show that the points ( iˆ ˆj+ 3 kˆ ) and 3( iˆ+ ˆj+ kˆ ) are equidistant from the plane r(5iˆ+ ˆj 7 kˆ ) + 9= 0 and lies on opposite side of it 6 AB= 3 iˆ ˆj+ kˆ and CD = 3iˆ+ ˆj+ 4kˆare two vectors The position vectors of the points A and C are 6iˆ+ 7ˆj+ 4kˆ and 9ˆj+ kˆ, respectively Find the position vector of a point P on the line AB and a point Q on the line CD such that PQ is perpendicular to AB and CD both 7 Show that the straight lines whose direction cosines are given by l + m n = 0 and mn + nl + lm = 0 are at right angles 8 If l 1, m 1, n 1 ; l, m, n ; l 3, m 3, n 3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l 1 + l + l 3, m 1 + m + m 3, n 1 + n + n 3 makes equal angles with them Objective Type Questions Choose the correct answer from the given four options in each of the Exercises from 9 to 36 9 Distance of the point (α,β,γ) from y-axis is

19 38 MATHEMATICS (A) β (B) β (C) β + γ (D) α + γ 30 If the directions cosines of a line are k,k,k, then (A) k>0 (B) 0<k<1 (C) k=1 (D) k 1 3 or The distance of the plane r ˆ 3 ˆ 6 ˆ i + j k = 1 from the origin is (A) 1 (B) 7 (C) 1 7 (D) None of these 3 The sine of the angle between the straight line plane x y + z = 5 is x y 3 z and the (A) (B) 4 5 (C) The reflection of the point (α,β,γ) in the xy plane is (A) (α,β,0) (B) (0,0,γ) (C) ( α, β,γ) (D) (α,β, γ) 34 The area of the quadrilateral ABCD, where A(0,4,1), B (, 3, 1), C(4, 5, 0) and D (, 6, ), is equal to (A) 9 sq units (B) 18 sq units (C) 7 sq units (D) 81 sq units 35 The locus represented by xy + yz = 0 is (A) A pair of perpendicular lines (B) A pair of parallel lines (C) A pair of parallel planes (D) A pair of perpendicular planes 36 The plane x 3y + 6z 11 = 0 makes an angle sin 1 (α) with x-axis The value of α is equal to (D) 10 (A) 3 (B) 3 (C) 7 (D) 3 7

20 THREE DIMENSIONAL GEOMETRY 39 Fill in the blanks in each of the Exercises 37 to A plane passes through the points (,0,0) (0,3,0) and (0,0,4) The equation of plane is 38 The direction cosines of the vector (iˆ ˆj k ˆ) are 39 The vector equation of the line x 5 y 4 z is 40 The vector equation of the line through the points (3,4, 7) and (1, 1,6) is 41 r The cartesian equation of the plane ( iˆ ˆj k ˆ) is State True or False for the statements in each of the Exercises 4 to 49 4 The unit vector normal to the plane x + y +3z 6 = 0 is 1 3 i ˆ ˆ j k ˆ The intercepts made by the plane x 3y + 5z +4 = 0 on the co-ordinate axis are, 4, The angle between the line r (5 iˆ ˆj 4 kˆ) ( iˆ ˆj kˆ) and the plane r(3 iˆ 4 ˆj k ˆ) 5 0 is 1 5 sin The angle between the planes r( iˆ 3 ˆj k ˆ) 1 and r( iˆ ˆj ) 4 is 1 5 cos The line r iˆ 3 ˆj kˆ ( iˆ ˆj kˆ) lies in the plane r(3 iˆ ˆj k ˆ) 0 47 The vector equation of the line x 5 y 4 z is

21 40 MATHEMATICS r 5 iˆ 4ˆj 6 kˆ (3iˆ 7ˆj kˆ) 48 The equation of a line, which is parallel to iˆ ˆj 3k ˆ and which passes through x 5 y z 4 the point (5,,4), is If the foot of perpendicular drawn from the origin to a plane is (5, 3, ), then the equation of plane is r(5 iˆ 3ˆj kˆ ) = 38

STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

Class 11 Maths Chapter 10. Straight Lines

1 P a g e Class 11 Maths Chapter 10. Straight Lines Coordinate Geometry The branch of Mathematics in which geometrical problem are solved through algebra by using the coordinate system, is known as coordinate

Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

VECTOR NAME OF THE CHAPTER PART-C FIVE MARKS QUESTIONS PART-E TWO OR FOUR QUESTIONS PART-A ONE MARKS QUESTIONS

NAME OF THE CHAPTER VECTOR PART-A ONE MARKS QUESTIONS PART-B TWO MARKS QUESTIONS PART-C FIVE MARKS QUESTIONS PART-D SIX OR FOUR MARKS QUESTIONS PART-E TWO OR FOUR QUESTIONS 1 1 1 1 1 16 TOTAL MARKS ALLOTED

C relative to O being abc,, respectively, then b a c.

2 EP-Program - Strisuksa School - Roi-et Math : Vectors Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A

COORDINATE GEOMETRY. 13. If a line makes intercepts a and b on the coordinate axes, then its equation is x y = 1. a b

NOTES COORDINATE GEOMETRY 1. If A(x 1, y 1 ) and B(x, y ) are two points then AB = (x x ) (y y ) 1 1 mx nx1 my ny1. If P divides AB in the ratio m : n then P =, m n m n. x1 x y1 y 3. Mix point of AB is,.

www.sakshieducation.com

LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c

NCERT. The coordinates of the mid-point of the line segment joining the points P (x 1. which is non zero unless the points A, B and C are collinear.

COORDINATE GEOMETRY CHAPTER 7 (A) Main Concepts and Results Distance Formula, Section Formula, Area of a Triangle. The distance between two points P (x 1 ) and Q (x, y ) is ( x x ) + ( y y ) 1 1 The distance

CHAPTER 4 PAIR OF STRAIGHT LINES

CHAPTER 4 PAIR OF STRAIGHT LINES TOPICS: 1. Equation of a pair of lines passing through the origin.angle between pair of lines 3.Bisectors of the angles between two lines. 4. pair of bisectors of angles

6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:

6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an inner-product space. This is the usual n- dimensional

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1

5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in

Announcements 2-D Vector Addition Today s Objectives Understand the difference between scalars and vectors Resolve a 2-D vector into components Perform vector operations Class Activities Applications Scalar

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.

VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position

7. ANALYTICAL GEOMETRY

7. ANALYTICAL GEOMETRY 7.1 Cartesian coordinate system: a. Number line We have studied the number line in our lower classes. On the number line, distances from a fixed point (called the origin or the zero

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.

REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described

HYPERBOLIC TRIGONOMETRY

HYPERBOLIC TRIGONOMETRY STANISLAV JABUKA Consider a hyperbolic triangle ABC with angle measures A) = α, B) = β and C) = γ. The purpose of this note is to develop formulae that allow for a computation of

9.3. Direction Ratios and Direction Cosines. Introduction. Prerequisites. Learning Outcomes. Learning Style

Direction Ratios and Direction Cosines 9.3 Introduction Direction ratios provide a convenient way of specifying the direction of a line in three dimensional space. Direction cosines are the cosines of

THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

16. Let S denote the set of positive integers 18. Thus S = {1, 2,..., 18}. How many subsets of S have sum greater than 85? (You may use symbols such

æ. Simplify 2 + 3 + 4. 2. A quart of liquid contains 0% alcohol, and another 3-quart bottle full of liquid contains 30% alcohol. They are mixed together. What is the percentage of alcohol in the mixture?

Vectors What are Vectors? which measures how far the vector reaches in each direction, i.e. (x, y, z).

1 1. What are Vectors? A vector is a directed line segment. A vector can be described in two ways: Component form Magnitude and Direction which measures how far the vector reaches in each direction, i.e.

Modern Geometry Homework.

Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

MATHEMATICS CLASS - XII BLUE PRINT - II. (1 Mark) (4 Marks) (6 Marks)

BLUE PRINT - II MATHEMATICS CLASS - XII S.No. Topic VSA SA LA TOTAL ( Mark) (4 Marks) (6 Marks). (a) Relations and Functions 4 () 6 () 0 () (b) Inverse trigonometric Functions. (a) Matrices Determinants

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

Mathematics 1. Lecture 5. Pattarawit Polpinit

Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance

STRAIGHT LINE COORDINATE GEOMETRY

STRAIGHT LINE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) The points P and Q have coordinates ( 7,3 ) and ( 5,0), respectively. a) Determine an equation for the straight line PQ, giving the answer

PROBLEMS 04 - PARABOLA Page 1

PROBLEMS 0 - PARABOLA Page 1 ( 1 ) Find the co-ordines of the focus, length of the lus-rectum and equion of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line x k 0 is a tangent

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

CHAPTER FIVE. 5. Equations of Lines in R 3

118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

Revision Coordinate Geometry. Question 1: The gradient of the line joining the points V (k, 8) and W (2, k +1) is is.

Question 1: 2 The gradient of the line joining the points V (k, 8) and W (2, k +1) is is. 3 a) Find the value of k. b) Find the length of VW. Question 2: A straight line passes through A( 2, 3) and B(

Mathematics Notes for Class 12 chapter 10. Vector Algebra

1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

CARTESIAN VECTORS AND THEIR ADDITION & SUBTRACTION

CARTESIAN VECTORS AND THEIR ADDITION & SUBTRACTION Today s Objectives: Students will be able to: a) Represent a 3-D vector in a Cartesian coordinate system. b) Find the magnitude and coordinate angles

Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

Section 12.1 Translations and Rotations

Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry (meaning equal measure ). In this section, we will investigate two types of isometries: translations

of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 459-2058 Mobile: (949) 510-8153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:

Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

Notes on spherical geometry

Notes on spherical geometry Math 130 Course web site: www.courses.fas.harvard.edu/5811 This handout covers some spherical trigonometry (following yan s exposition) and the topic of dual triangles. 1. Some

CHAPTER 38 INTRODUCTION TO TRIGONOMETRY

CHAPTER 38 INTRODUCTION TO TRIGONOMETRY EXERCISE 58 Page 47. Find the length of side x. By Pythagoras s theorem, 4 = x + 40 from which, x = 4 40 and x = 4 40 = 9 cm. Find the length of side x. By Pythagoras

CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

Geometry Unit 1. Basics of Geometry

Geometry Unit 1 Basics of Geometry Using inductive reasoning - Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture- an unproven statement that is based

Lesson 13: Proofs in Geometry

211 Lesson 13: Proofs in Geometry Beginning with this lesson and continuing for the next few lessons, we will explore the role of proofs and counterexamples in geometry. To begin, recall the Pythagorean

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t

Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are

MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

a a. θ = cos 1 a b ) b For non-zero vectors a and b, then the component of b along a is given as comp

Textbook Assignment 4 Your Name: LAST NAME, FIRST NAME (YOUR STUDENT ID: XXXX) Your Instructors Name: Prof. FIRST NAME LAST NAME YOUR SECTION: MATH 0300 XX Due Date: NAME OF DAY, MONTH DAY, YEAR. SECTION

Assignment 3. Solutions. Problems. February 22.

Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES This section refers to the properties of straight lines and curves using rules found by the use of cartesian co-ordinates. The Gradient of a Line. As

POINT OF INTERSECTION OF TWO STRAIGHT LINES

POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel

1916 New South Wales Leaving Certificate

1916 New South Wales Leaving Certificate Typeset by AMS-TEX New South Wales Department of Education PAPER I Time Allowed: Hours 1. Prove that the radical axes of three circles taken in pairs are concurrent.

Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

206 MATHEMATICS CIRCLES

206 MATHEMATICS 10.1 Introduction 10 You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have

Chapter 1: Points, Lines, Planes, and Angles

Chapter 1: Points, Lines, Planes, and Angles (page 1) 1-1: A Game and Some Geometry (page 1) In the figure below, you see five points: A,B,C,D, and E. Use a centimeter ruler to find the requested distances.

Pythagorean theorems in the alpha plane

MATHEMATICAL COMMUNICATIONS 211 Math. Commun., Vol. 14, No. 2, pp. 211-221 (2009) Pythagorean theorems in the alpha plane Harun Baris Çolakoğlu 1,, Özcan Gelişgen1 and Rüstem Kaya 1 1 Department of Mathematics

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications

C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

Laboratory 2 Application of Trigonometry in Engineering

Name: Grade: /26 Section Number: Laboratory 2 Application of Trigonometry in Engineering 2.1 Laboratory Objective The objective of this laboratory is to learn basic trigonometric functions, conversion

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

Math 366 Lecture Notes Section 11.1 Basic Notions (of Geometry)

Math 366 Lecture Notes Section. Basic Notions (of Geometry) The fundamental building blocks of geometry are points, lines, and planes. These terms are not formally defined, but are described intuitively.

Announcements. Moment of a Force

Announcements Test observations Units Significant figures Position vectors Moment of a Force Today s Objectives Understand and define Moment Determine moments of a force in 2-D and 3-D cases Moment of

APPLICATION OF DERIVATIVES

6. Overview 6.. Rate of change of quantities For the function y f (x), d (f (x)) represents the rate of change of y with respect to x. dx Thus if s represents the distance and t the time, then ds represents

Formal Geometry S1 (#2215)

Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following course: Formal Geometry S1 (#2215)

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

5 Hyperbolic Triangle Geometry

5 Hyperbolic Triangle Geometry 5.1 Hyperbolic trigonometry Figure 5.1: The trigonometry of the right triangle. Theorem 5.1 (The hyperbolic right triangle). The sides and angles of any hyperbolic right

Year 12 Pure Mathematics. C1 Coordinate Geometry 1. Edexcel Examination Board (UK)

Year 1 Pure Mathematics C1 Coordinate Geometry 1 Edexcel Examination Board (UK) Book used with this handout is Heinemann Modular Mathematics for Edexcel AS and A-Level, Core Mathematics 1 (004 edition).

acute angle acute triangle Cartesian coordinate system concave polygon congruent figures

acute angle acute triangle Cartesian coordinate system concave polygon congruent figures convex polygon coordinate grid coordinates dilatation equilateral triangle horizontal axis intersecting lines isosceles

CHAPTER 1. LINES AND PLANES IN SPACE

CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given

Time: 1 hour 10 minutes

[C00/SQP48] Higher Time: hour 0 minutes Mathematics Units, and 3 Paper (Non-calculator) Specimen Question Paper (Revised) for use in and after 004 NATIONAL QUALIFICATIONS Read Carefully Calculators may

BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

CONSTRUCTIONS MODULE - 3 OBJECTIVES. Constructions. Geometry. Notes

MODULE - 3 Constructions 18 CONSTRUCTIONS One of the aims of studying is to acquire the skill of drawing figures accurately. You have learnt how to construct geometrical figures namely triangles, squares

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

FORCE VECTORS. Lecture s Objectives

CHAPTER Engineering Mechanics: Statics FORCE VECTORS Tenth Edition College of Engineering Department of Mechanical Engineering 2b by Dr. Ibrahim A. Assakkaf SPRING 2007 ENES 110 Statics Department of Mechanical

M243. Fall Homework 2. Solutions.

M43. Fall 011. Homework. s. H.1 Given a cube ABCDA 1 B 1 C 1 D 1, with sides AA 1, BB 1, CC 1 and DD 1 being parallel (can think of them as vertical ). (i) Find the angle between diagonal AC 1 of a cube

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: School Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, June 16, 2009 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name of

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

Patterns, Equations, and Graphs. Section 1-9

Patterns, Equations, and Graphs Section 1-9 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate

NATIONAL SENI CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 00 MEMANDUM MARKS: 0 This memorandum consists of pages. Copyright reserved Mathematics/P DoE/Feb. March 00 QUESTION. Range 6 maimum and minimum

Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

Chapter One. Points, Lines, Planes, and Angles

Chapter One Points, Lines, Planes, and Angles Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately

Chapter 1. Foundations of Geometry: Points, Lines, and Planes

Chapter 1 Foundations of Geometry: Points, Lines, and Planes Objectives(Goals) Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in