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

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1 VECTOR ALGEBRA Chapter 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 vector of a point P (x, y, z) is given as OP = xiˆ+ y ˆj+ zkˆ magnitude as OP = x + y + z, where O is the origin and its 1014 The scalar components of a vector are its direction ratios, and represent its projections along the respective axes 1015 The magnitude r, direction ratios (a,, c) and direction cosines (l, m, n) of any vector are related as: a c l=, m=, n= r r r 1016 The sum of the vectors representing the three sides of a triangle taken in order is The triangle law of vector addition states that If two vectors are represented y two sides of a triangle taken in order, then their sum or resultant is given y the third side taken in opposite order 1018 Scalar multiplication If a is a given vector and λ a scalar, then λ a is a vector whose magnitude is λ a = λ a The direction of λ a is same as that of a if λ is positive and, opposite to that of a if λ is negative

2 VECTOR ALGEBRA Vector joining two points If P 1 (x 1, y 1,z 1 ) and P (x, y,z ) are any two points, then PP = ( x x ) iˆ+ ( y y ) ˆj+ ( z z ) kˆ PP = ( x x ) + ( y y ) + ( z z ) Section formula The position vector of a point R dividing the line segment joining the points P and Q whose position vectors are a and (i) in the ratio m : n internally, is given y na m m n (ii) in the ratio m : n externally, is given y m na m n Projection of a along is a is 1011 Scalar or dot product a and the Projection vector of a along The scalar or dot product of two given vectors a and having an angle θ etween them is defined as a = a cos θ 1011 Vector or cross product The cross product of two vectors a and having angle θ etween them is given as a = a sin θ ˆn,

3 06 MATHEMATICS where ˆn is a unit vector perpendicular to the plane containing a and and a,, ˆn form a right handed system If a = ai 1 ˆ+ a ˆ ˆj + ak and = i ˆ ˆ 1 + ˆ j+ k are two vectors and λ is any scalar, then a + = ( a1 + ˆ ˆ ˆ 1) i + ( a + ) j+ ( a+ ) k λ a = (λ a ˆ ˆ ˆ 1) i + (λ a) j+ (λ a) k a = a a + a iˆ ˆj kˆ a a1 1 c1 = a c = ( 1 c c 1 ) î + (a c 1 c 1 c ) ĵ + (a 1 a 1 ) ˆk Angle etween two vectors a and is given y cos θ = a a = a+ a + a 1 1 a + a + a Solved Examples Short Answer (SA) Example 1 Find the unit vector in the direction of the sum of the vectors a = iˆ ˆj+ kˆ and = iˆ+ ˆj+ kˆ Solution Let c denote the sum of a and We have c = ( iˆ ˆj+ kˆ) + ( iˆ+ ˆj+ kˆ) = iˆ + 5 kˆ Now c = = 6

4 VECTOR ALGEBRA 07 Thus, the required unit vector is c c= = 1 ( i+ 5k) = 1 i + 5 k c Example Find a vector of magnitude 11 in the direction opposite to that of PQ, where P and Q are the points (1,, ) and ( 1, 0, 8), respetively Solution The vector with initial point P (1,, ) and terminal point Q ( 1, 0, 8) is given y PQ = ( 1 1) î + (0 ) ĵ + (8 ) ˆk = î ĵ + 6 ˆk Thus Q P = PQ = iˆ+ ˆj 6kˆ QP = + + = + + = = ( 6) Therefore, unit vector in the direction of QP is given y QP Q P Q P iˆ ˆj 6kˆ 7 Hence, the required vector of magnitude 11 in direction of QP is 11 QP = 11 iˆ ˆj 6k ˆ 7 = iˆ+ ˆ j 66 kˆ Example Find the position vector of a point R which divides the line joining the two points P and Q with position vectors OP a and OQ a, respectively, in the ratio 1:, (i) internally and (ii) externally Solution (i) The position vector of the point R dividing the join of P and Q internally in the ratio 1: is given y OR ( a ) 1( a ) 5a 1

5 08 MATHEMATICS (ii) The position vector of the point R dividing the join of P and Q in the ratio 1 : externally is given y ( a+ ) 1( a ) OR = = a Example 4 If the points ( 1, 1, ), (, m, 5) and (,11, 6) are collinear, find the value of m Solution Let the given points e A ( 1, 1, ), B (, m, 5) and C (, 11, 6) Then A B = ( + 1) iˆ+ ( m+ 1) ˆj+ (5 ) kˆ = iˆ+ ( m+ 1) ˆj + kˆ and A C = ( + 1) iˆ+ (11+ 1) ˆj+ (6 ) kˆ = 4iˆ+ 1 ˆj + 4kˆ Since A, B, C, are collinear, we have AB = λ A C, ie, ( iˆ ( m 1) ˆj kˆ) λ (4 iˆ+1 ˆj+ 4 kˆ) = 4 λ and m + 1 = 1 λ Therefore m = 8 Example 5 Find a vector r of magnitude units which makes an angle of π 4 and π with y and z - axes, respectively Solution Here m = π 1 cos = and n = cos π 4 = 0 Therefore, l + m + n = 1 gives l = 1 l = ± 1

6 VECTOR ALGEBRA 09 Hence, the required vector r = ( liˆ+ m ˆj + nkˆ ) is given y r = 1 1 ( ˆ ˆ 0 ˆ) i j k = r = ± i ˆ + ˆ j Example 6 If a = iˆ ˆj + kˆ, = î + ĵ k ˆ and c = î + j ˆ ˆk, find λ such that a is perpendicular to c Solution We have λ + c = λ ( î + ĵ ˆk ) + ( î + ĵ ˆk ) Since a (λ + c ), a (λ + c ) = 0 = ( λ + 1) î + (λ + ) ĵ (λ + 1) ˆk ( î ĵ + ˆk ) [( λ + 1) î + (λ + ) ĵ (λ + 1) ˆk ] = 0 (λ + 1) (λ + ) (λ + 1) = 0 λ = Example 7 Find all vectors of magnitude 10 that are perpendicular to the plane of iˆ ˆj k ˆ and iˆ ˆj 4k ˆ Solution Let a = iˆ ˆj k ˆ and = iˆ ˆj 4k ˆ Then iˆ ˆj kˆ a 1 1 iˆ(8 ) ˆj(4 1) k ˆ( ) 1 4 = 5 î 5 ĵ + 5 ˆk a (5) ( 5) (5) (5) 5

7 10 MATHEMATICS Therefore, unit vector perpendicular to the plane of a and is given y a 5iˆ 5ˆj 5kˆ a 5 Hence, vectors of magnitude of 10 that are perpendicular to plane of a and are 10 5iˆ 5ˆj 5k ˆ 5, ie, 10( iˆ ˆj k ˆ) Long Answer (LA) Example 8 Using vectors, prove that cos (A B) = cosa cosb + sina sinb Solution Let OP and OQ e unit vectors making angles A and B, respectively, with positive direction of x-axis Then QOP = A B [Fig 101] We know OP = OM + MP iˆcosa + ˆjsin A and OQ = ON + NQ iˆcos B + ˆjsin B By definition OP OQ OP OQ cos A-B = cos (A B) (1) OP 1 OQ In terms of components, we have OP OQ = (cosa i ˆ ˆ jsina)(cosb i ˆ ˆ jsinb) = cosa cosb + sina sinb () From (1) and (), we get cos (A B) = cosa cosb + sina sinb

8 VECTOR ALGEBRA 11 sin A sin B sin C Example 9 Prove that in a Δ ABC,, where a,, c represent the a c magnitudes of the sides opposite to vertices A, B, C, respectively Solution Let the three sides of the triangle BC, CA and AB e represented y a, andc, respectively [Fig 10] We have a c 0 ie, a c which pre cross multiplying y a, and post cross multiplying y, gives a = c a and a c respectively Therefore, a c c a a c c a asin ( C) csin ( A) ca sin ( B) a sin C = c sina = ca sinb Dividing y ac, we get sin C sin A sin B c a Ojective Type Questions ie sin A sin B sin C a c Choose the correct answer from the given four options in each of the Examples 10 to 1 Example 10 The magnitude of the vector 6iˆ ˆj k ˆ is

9 1 MATHEMATICS 5 (B) 7 (C) 1 (D) 1 Solution (B) is the correct answer Example 11 The position vector of the point which divides the join of points with position vectors a and a in the ratio 1 : is a (B) a (C) 5a (D) 4 a Solution (D) is the correct answer Applying section formula the position vector of the required point is ( a ) 1( a ) 4a 1 Example 1 The vector with initial point P (,, 5) and terminal point Q(, 4, 7) is iˆ ˆj k ˆ (B) 5iˆ 7ˆj 1kˆ (C) iˆ ˆj k ˆ (D) None of these Solution is the correct answer Example 1 The angle etween the vectors iˆ ˆj and ĵ k ˆ is (B) (C) Solution (B) is the correct answer Apply the formula cosθ = (D) a a 5 6 Example 14 The value of λ for which the two vectors iˆ ˆj k ˆ and iˆ ˆj kˆ are perpendicular is (B) 4 (C) 6 (D) 8 Solution (D) is the correct answer

10 VECTOR ALGEBRA 1 Example 15 The area of the parallelogram whose adjacent sides are iˆ ˆj k ˆ is î k ˆ and (B) (C) (D) 4 Solution (B) is the correct answer Area of the parallelogram whose adjacent sides are a and is a ˆ Example 16 If a = 8, and a 1, then value of a is 6 (B) 8 (C) 1 (D) None of these Solution (C) is the correct answer Using the formula a a sinθ, we get π θ=± 6 Therefore, a = a cos = 8 = 1 Example 17 The vectors ˆj+ kˆ and iˆ ˆj+ 4kˆ represents the two sides AB and AC, respectively of a ΔABC The length of the median through A is 4 (B) 48 (C) 18 (D) None of these Solution is the correct answer Median AD is given y 1 ˆ ˆ ˆ 4 AD = i+ j+ 5k = Example 18 The projection of vector a ˆ i ˆj kˆ along iˆ ˆj kˆ is

11 14 MATHEMATICS (B) 1 (C) (D) 6 Solution is the correct answer Projection of a vector a on is a = ( iˆ ˆj kˆ)( iˆ ˆj kˆ) = Example 19 If aand are unit vectors, then what is the angle etween a to e a unit vector? aand for 0 (B) 45 (C) 60 (D) 90 Solution is the correct answer We have ( a ) a a a = cosθ = θ = 0 Example 0 The unit vector perpendicular to the vectors iˆ ˆj and iˆ ˆj forming a right handed system is ˆk (B) ˆk (C) iˆ ˆj (D) iˆ ˆj Solution is the correct answer Required unit vector is iˆ ˆj iˆ ˆj iˆ ˆj iˆ ˆj = kˆ k ˆ Example 1 If a and 1 k, then ka lies in the interval [0, 6] (B) [, 6] (C) [, 6] (D) [1, ]

12 VECTOR ALGEBRA 15 Solution is the correct answer The smallest value of ka will exist at numerically smallest value of k, ie, at k = 0, which gives ka k a 0 0 The numerically greatest value of k is at which ka 6 10 EXERCISE Short Answer (SA) 1 Find the unit vector in the direction of sum of vectors a ˆ i ˆj kˆ and ˆj kˆ If a iˆ ˆj kˆ and iˆ ˆj kˆ, find the unit vector in the direction of (i) 6 (ii) a Find a unit vector in the direction of PQ, where P and Q have co-ordinates (5, 0, 8) and (,, ), respectively 4 If aand are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 15 BA 5 Using vectors, find the value of k such that the points (k, 10, ), (1, 1, ) and (, 5, ) are collinear 6 A vector r is inclined at equal angles to the three axes If the magnitude of r is units, find r 7 A vector r has magnitude 14 and direction ratios,, 6 Find the direction cosines and components of r, given that r makes an acute angle with x-axis 8 Find a vector of magnitude 6, which is perpendicular to oth the vectors iˆ ˆj kˆ and 4 iˆ ˆj k ˆ 9 Find the angle etween the vectors iˆ ˆj k ˆ and iˆ 4ˆj k ˆ 10 If a c 0, show that a c c a Interpret the result geometrically? 11 Find the sine of the angle etween the vectors a iˆ ˆj kˆ iˆ ˆj 4kˆ and

13 16 MATHEMATICS 1 If A, B, C, D are the points with position vectors iˆ ˆj k ˆ, iˆ ˆj k ˆ, iˆ kˆ,iˆ ˆj k, ˆ respectively, find the projection of AB along CD 1 Using vectors, find the area of the triangle ABC with vertices A(1,, ), B(, 1, 4) and C(4, 5, 1) 14 Using vectors, prove that the parallelogram on the same ase and etween the same parallels are equal in area Long Answer (LA) c a 15 Prove that in any triangle ABC, cos A, where a,, c are the c magnitudes of the sides opposite to the vertices A, B, C, respectively 16 If a,, c determine the vertices of a triangle, show that 1 c c a a gives the vector area of the triangle Hence deduce the condition that the three points a,, c are collinear Also find the unit vector normal to the plane of the triangle 17 Show that area of the parallelogram whose diagonals are given y a and is a Also find the area of the parallelogram whose diagonals are iˆ ˆj kˆ and iˆ ˆj k ˆ 18 If a = iˆ ˆj k ˆ and ˆj kˆ, find a vector c such that a c and ac Ojective Type Questions Choose the correct answer from the given four options in each of the Exercises from 19 to (MCQ) 19 The vector in the direction of the vector iˆ ˆj kˆ that has magnitude 9 is (C) iˆ ˆj k ˆ (B) iˆ ˆj kˆ ( iˆ ˆj k ˆ) (D) 9( iˆ ˆj kˆ )

14 VECTOR ALGEBRA 17 0 The position vector of the point which divides the join of points a and a in the ratio : 1 is a (B) 7a 8 4 (C) a 4 1 The vector having initial and terminal points as (, 5, 0) and (, 7, 4), respectively is (D) 5a 4 iˆ 1 ˆj 4k ˆ (B) 5iˆ ˆj 4kˆ (C) 5iˆ ˆj 4k ˆ (D) iˆ ˆj kˆ The angle etween two vectors a and with magnitudes and 4, respectively, and a is 6 (B) (C) Find the value of λ such that the vectors a ˆ i ˆj kˆ orthogonal (D) 5 and iˆ ˆj kˆ are 0 (B) 1 (C) (D) 5 4 The value of λ for which the vectors iˆ 6ˆj kˆ and iˆ 4ˆj k ˆ are parallel is (B) 5 The vectors from origin to the points A and B are a iˆ ˆj kˆ and iˆ ˆj kˆ,respectively, then the area of triangle OAB is (C) 5 (D) 5 40 (B) 5 (C) 9 (D) 1 9

15 18 MATHEMATICS 6 For any vector a, the value of ˆ ( ) ( ˆ a i a j) ( a kˆ ) is equal to a (B) a (C) 4 a (D) a 7 If a = 10, = and a 1, then value of a 5 (B) 10 (C) 14 (D) 16 8 The vectors iˆ ˆj kˆ, iˆ ˆj kˆ and iˆ ˆj kare ˆ coplanar if λ = (B) λ = 0 (C) λ = 1 (D) λ = 1 9 If a,, c are unit vectors such that a c 0, then the value of a c ca is is 1 (B) (C) (D) None of these 0 Projection vector of a on is a (B) a (C) a (D) a ˆ a a 1 If a,, c are three vectors such that a c 0 and a,, c 5, then value of a c ca is 0 (B) 1 (C) 19 (D) 8 If a 4 and, then the range of a is [0, 8] (B) [ 1, 8] (C) [0, 1] (D) [8, 1] The numer of vectors of unit length perpendicular to the vectors a= iˆ+ ˆj+ kˆ and = ˆj+ kˆ is one (B) two (C) three (D) infinite Fill in the lanks in each of the Exercises from 4 to 40 4 The vector a + isects the angle etween the non-collinear vectors a and if

16 VECTOR ALGEBRA 19 5 If ra 0, r 0, and rc 0 for some non-zero vector r, then the value of a( c) is 6 The vectors a i j kˆ and i k are the adjacent sides of a parallelogram The acute angle etween its diagonals is 7 The values of k for which are 8 The value of the expression 1 ka a and ka a is parallel to a holds true a + ( a ) is 9 If a a = 144 and a 4, then is equal to 40 If a is any non-zero vector, then ˆˆ ( ) ˆˆ ai i a j j ak ˆ kˆ equals State True or False in each of the following Exercises 41 If a, then necessarily it implies a 4 Position vector of a point P is a vector whose initial point is origin 4 If a a, then the vectors a and are orthogonal 44 The formula ( a ) a a is valid for non-zero vectors a and 45 If a and are adjacent sides of a rhomus, then a = 0