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

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1 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. (x, y, z). Length of line segment and which direction the vector is headed, denoted by an arrow. They may be typed in bold such as v, or as a letter with a line or arrow above or below it, such as, or. A vector the same length as v but going in the opposite direction is written as. Two vectors are equal if and only if their lengths and directions are the same. Equivalently, if their components are equal. Numerical representation of vectors v Magnitude and direction form This vector is written as ( ) or ( ) The first representation is called a column vector, and the second is a row vector. To convert between the two forms we use Pythagoras Theorem and trigonometry. Example 1 Find the magnitude and direction of Magnitude is 2 b 4 = 4.47 to 2 decimal places To find the angle we use the tangent function on the components. 4 tan 2 2 to 2 d.p. 2 So the vector b 4 is equivalent to a line of length 4.47 at an angle of to the horizontal, measured counterclockwise.

2 Unit Vector The notation â (known as hat ) is used for unit vectors. These are a special type of vector whose magnitude is 1 unit. Example 2 Find a unit vector in the same direction as a = (3, 0, - 4). Find the magnitude a ( ) = 5 Unit Vector ( ) Divide each component by the magnitude. Zero Vector The vector which has no magnitude and no direction is known as the zero vector and is denoted 0. The zero vector is (0, 0, 0). It is the only vector with zero magnitude. Direction Cosines For vector p = ( ), let represent the angles between the axes respectively. are the direction cosines Example 3 Find the direction cosines for the vector r = ( ) Find the magnitude ( ) The modulus of a vector is another name for its magnitude. In other words it is the length of the vector. 1

3 Exercise 1 1. Convert the following vectors into magnitude and direction: 3 a 1 2 b 6 c d 2 2. Find the magnitude of the vectors u = (3, -4, 5) v = ( 1, -2, 4) w = (-2, 0, -3) x = ( -4, -5, 1) 3. Find values for a and b that make u = (2, 3+a, 5 b) and v = ( b-1, 2, 1-a) equal. 4. Find unit vectors in the direction of each vector u = (2, -4, 4) v = ( -2, -6, 3) w = (0, 8, -6) x = ( -2, -2, 0) 5. Find the direction cosines for each vector u = (4, 7, -4) v = ( -1, -1, 2) w = (3, 2, 1) 2

4 2. Arithmetic operations on vectors Addition and Subtraction of vectors Add or subtract each corresponding component together. ( ) ( ) ( ) ( ) ( ) ( ) Example 1 ( ) ( ) ( ) The addition of vectors obeys the commutative and associative laws. Example 2 ( ) ( ) ( ) Multiplication of a vector by a scalar Multiplying a vector by a scalar has the effect of multiplying every component by that value. ( ) ( ) Example 3 ( ) ( ) This is the equivalent of taking 3 copies of ( ) and placing them end to end. In magnitude/direction form this is equivalent to multiplying the magnitude by the scalar. If the scalar is negative then the angle is rotated by180. The multiplication of a vector by a scalar obeys the distributive and associative laws. 3

5 Unit Vectors i, j, k i = (1, 0, 0) j = (0, 1, 0) k = (0, 0, 1) Any 3-dimensional vector (i.e. a vector with three components) can be written in terms of i, j, k. u = (a, b, c) = ai + bj + ck v = (-2, 5, -3) = -2i + 5j 3k Position Vectors Position vectors are vectors giving the position of a point, relative to a fixed point (eg. the origin). The point A(4,5,-3) has position vector ( ) where O is the origin. Example 4 Find the ratio that AB:AC for the points A(15, -10, -8) B(5, 1, 0) C(-15, 23, 16) ( ( ) ( ) ) ( ) ( ( )) ( ) = 3 ( ) ( ) hence AB:AC = 1:3 Parallel Vectors Vectors p and q are parallel if and only if p = tq for some t Example 5 Find values for a and b that make p = (3, b, 6) and q = (a, -8, 4) parallel. To be parallel in the z components Consider the x components Consider the y components 4

6 Exercise 2 1. If a = ( ) b = ( ) and c = ( ) Find the components of i. a+ b c ii. c - b iii. c a b iv. 2a + 3b v. 2b 5a vi. a 2b + c 2. Using the point R(4,3) and vector u = ( ) find the position of the points A = R - B = R + C = R - D = R + 3. If p = i 2j + 2k and q = 3i + 2j 6k calculate a. p b. q c. p + q d. p - q 4. For the points A(3, 6, 5) B(7, 8, 1) and C(15, 12, -7) a. Show that the points A, B and C are collinear. b. Find the ratio AB:BC 5. For the points P(4, -4, 6) Q (5, -6, 5) R(2, 0, 8) a. Show that the points P, Q and R are collinear. b. Find the ratio PQ:QR 6. Find values for x, and y that make p = ( ) q =( ) parallel. 5

7 3. Scalar (dot) product The most common use for the scalar product is to find the angle between two vectors. In particular, it is used to find out if two vectors are at right-angles to each other. If two vectors are at right-angles to each other then they are called perpendicular or orthogonal. The formula for the scalar product is: where is the angle between the vectors. Also is used as shorthand for n i 1 ab Where a i and b i are the components of a and b i i Examples 1. Given a ( 3, 4), b ( 1, 2) find a b. a b Given a ( 2, 31, ), b ( 1, 2, 4) find a b ( ) ( ) ( ). Comment on your result. As a b 0, this means that the vectors are perpendicular to each other. 3. Find the angle between a ( 52, ) and b ( 2, 3). a b a b ( 3 ) cos The angle between a and b is approximately degrees. 6

8 Exercise 3 1 Find the value of a b for the following pairs of vectors: a) a ( 2, 5), b ( 4, 6) b) a ( 1, 2, 4), b ( 2, 9, 3) c) a ( 3, 2, 7), b ( 1, 4, 9) 2 Use the scalar product to find the angle between the vectors, rounding your answers to 2d.p. a) u = (1, 2, 3) v = (0, -2, -1) b) a ( 3, 2, 4), b ( 1, 6, 3) c) r = ( ) s = ( ) d) m = ( ) n = ( ) e) p = -2i + 3j + k e = 4i + 7j + 3k f) e = 2i - 2j + 6k f = 5i + 9j + 13k 3. Find the value of x if the vectors u = ( ) and v = ( ) are perpendicular. 4. Find the value of y if the vectors u = ( ) and v = ( ) are perpendicular. 5. Evaluate for the points A(3, 2, 1) B(4, 5, -2) C(4, 2, 3). 7

9 4. Vector (cross) product The vector or cross product is another method for multiplying vectors. Given vectors a and b it is used to find a vector c which is perpendicular to both vectors. It is used with 3-dimensional vectors. One way to express the cross product is: The product. between a and b is not a multiplication sign, it is a cross to denote cross The a i and b i represent the i th component of vectors a and b. The modulus symbol means we need to find the determinant. The calculation that is performed in order to find the cross product is this: iˆ ˆj kˆ a b a a a ( a b a b ) iˆ ( a b a b ) ˆj ( a b a b ) kˆ b b b Note that the symbol in front of the middle bracket is -. This will always be the case. Example Find a vector perpendicular to both a ( 2, 6, 5) and b ( 2, 1, 4). iˆ ˆj kˆ ( ) ( ) ˆ ( ) ( ) ˆ ( ) ( ) a b i j kˆ ( ) ( ) ( ) The vector ( 29, 2, 14) is perpendicular to a ( 2, 6, 5) and b ( 2, 1, 4). 8

10 Exercise 4 1 Find a vector perpendicular to each of the following pairs of vectors: a) a ( 1, 4, 2), b ( 3, 2, 7) b) a ( 4, 1, 3), b ( 2, 6, 4) c) a ( 2, 31, ), b ( 5, 2, 8) d) e = 3i - 4j + 5k f = -1i + 2j + 6k e) e = 2i + 3j - k f = -3i + 4j - 2k f) e = 8i - 5j + 2k f = -i + 3j + 2k 2. Find ( ) where a = ( ) b = ( ) and c = ( ) 9

11 Area of a Triangle Area of a Triangle is = Example If points A(1,3,2) B ( 4, 7, -2) and C (4, -2, -5) form a triangle as follows; B Hence vectors are b b = B A = (3, 4, -4) c = C A = (3, -5, -7) A θ c A = C = (-28 20)i (-21 (-12)j + (-15-12)k = -48 i - (-9) j + (-27) k = -48i + 9j - 27k A = = ( ) ( ) = = 27.9 Exercise 5 Find the area of the triangle formed by the points 1. A(1,1,-1) B (1, 0,3) C(-2,2,1) 2. X(1, 2, -4) Y(0, 8, 5) Z ( 3, -6, 4) 3. P(4, -4, 5) Q(3, -5, 5) R(3, -4, 4) 4. L(7, -5, 3) M(9, -6, 0) N(10, -10, 2) 10

12 Answers to Exercises Exercise 1 1. a magnitude= angle= b magnitude= angle= c magnitude= angle= d magnitude= angle= a = -1 b = 3 4. ( ) ( ) ( ) ( ) 5. u v w Exercise 2 1. i. ( ) ii ( ) iii ( ) iv ( ) v ( ) vi ( ) 2. A(-5, -3) B(13, 9) C(-8, -5) D(19, 13) 3. a. b. c. d. 4. a. b. 1:2 5. a. b. 1:3 6. Exercise 3 1. a) 22 b) -4 c) a) b) c) d) e) f)

13 Exercise 4 Vectors 1. a) ( 24, 1, 10) b) ( 14, 22, 26) c) ( 22, 21, 19) d) -34i - 23j + 2k e) -2i + 7j + 17k f) -16i - 18j + 19k Exercise These exercises were adapted from Mathematics Worksheets originally created by Study Advice Service at Hull University. Web: Many thanks for use of these materials. Any comments can be sent to the above or to 12

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