Vectors Summary A vector includes magnitude (size) and direction. Academic Skills Advice Types of vectors: Line vector: Free vector: Position vector: Unit vector (n ): can slide along the line of action. not restricted, defined by magnitude & direction but can be anywhere. one end is fixed and it usually starts at the origin ( O = 4i 5j, means start at the origin and go along 4 and down 5). The vector with a length of 1 (sometimes called the normalised vector). i is a unit vector in the x-direction j is a unit vector in the y-direction k is a unit vector in the k-direction Addition of Vectors: When adding vectors you can draw them as a chain with the 2 nd vector starting where the 1 st one ended. if A is the vector and C is the vector A C Adding them gives: A + C = AC A C The resultant vector When you are given the components you can just add the i s and add the j s: add the vectors: v = 2i + 4j and u = 5i + 2j v + u = 7i + 6j Magnitude of Vectors: To find the magnitude of vector a (denoted a ) you would use Pythagoras. if a = 6i 3j + 2k, find a a = 6 2 + ( 3) 2 + 2 2 = 49 = 7 Finding the unit Vector: The unit vector (n ) has a length of 1. So to find the unit vector (i.e. to make the length 1) we need to divide each component of the vector by the original length, i.e. n = n n find the unit vector (b ) of the vector b = 2i 3j + k Therefore: b = b = 2 2 + ( 3) 2 + 1 2 = 14 b = b b = (2, 3,1) 14 2 i 3 j + 1 k (and b = 1) 14 14 14 H Jackson 2014/15/16 / Academic Skills 1 Except where otherwise noted, this work is licensed under http://creativecommons.org/licenses/by-nc-sa/3.0/
Direction cosines: The direction cosines are the cosines of the angles between the vector and each of the 3 axes. For example: and i.e. If v = ai + bj + ck l = cos(α) where α is the angle between the vector and the x-axis, m = cos(β) where β is the angle between the vector and the y-axis n = cos(γ) where γ is the angle between the vector and the z-axis. The direction cosines (l, m and n) are: l = a m = b n = c (Remember that can be found by using Pythagoras: = a 2 + b 2 + c 2 ) let a = 3i 2j + 6k a = 3 2 + ( 2) 2 + 6 2 = 7 the direction cosines (l, m & n) are: l = 3 2, m =, 7 7 n = 6 7 (We have found that cos(α) = 3 and so the angle between the vector and the x-axis is: 7 α = cos 1 ( 3 ) = 7 64.60. We can find the other angles in the same way) Resolving Vectors: We can use basic trigonometry to resolve a vector into its x and y components. Remember that if you go: through the angle you use cos away from the angle you use sin (because we are considering hypotenuse and adjacent) (because we are considering hypotenuse and opposite) Vector a has an angle of 40 o and a length of 8. y x component: a x = 8cos40 = 6.13 40 8 a x a y x y component: a y = 8sin40 = 5.14 Vector Multiplication: There are 2 types of vector multiplication the dot (scalar) product and the cross (vector) product. See the following pages for explanations. H Jackson 2014/15/16 / Academic Skills 2
Dot (Scalar) Product: The dot product of the vectors a and b is written as a. b (a dot b) The result of the dot product is a scalar: i.e. vector. vector = scalar For two vectors a and b the dot product is calculated by multiplying the coefficients of i PLUS multiplying the coefficients of j PLUS multiplying the coefficients of k (you could think of it as: (i i) + (j j) + (k k)) a = 2i + 3j + 5k b = 4i + j + 6k (i s) (j s) (k s) a. b = (2 4) + (3 1) + (5 6) = 41 We have found that the dot product of the vectors a and b = 41 Finding the angle etween 2 vectors: The dot product can also be used to find the angle between two vectors using the following equation: a. b = a b cosθ We already know that a. b = 41 (above) and we can work out the right hand side as follows: a b = 2 2 + 3 2 + 5 2 4 2 + 1 2 + 6 2 = 38 53 Now we can use the equation to find the angle between the 2 vectors: a. b = a b cosθ 41 = 38 53 cosθ θ = cos 1 ( 41 38 53 ) θ = 24 o Parallel or perpendicular? If a. b = 0 then cosθ = 0 θ = 90 o perpendicular If a. b = ab then cosθ = 1 θ = 0 o parallel H Jackson 2014/15/16 / Academic Skills 3
Cross (Vector) Product: The cross product of the vectors a and b is written as a b (a cross b) The result of the cross product is a vector: i.e. vector vector = vector For two vectors a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k the cross product is calculated as follows: This looks like a complicated formula to remember but there is an easy way to do the cross product using a matrix. a = 2i + 4j + 3k b = i + 5j 2k Set up a matrix with a above b. i j k a b = 2 4 3 1 5 2 To find the i component, cover up everything in the i row & column, and find the determinant of what s left (the minor). Do the same for j and k and apply the alternate signs (see matrix summary sheet for more help). If you re not sure about (or just don t like) matrices see the final page for an easy way to remember, and write out, the formula. a b = + 4 3 3 4 i 2 j + 2 5 2 1 2 1 5 k = +((4 2) (3 5))i ((2 2) (3 1))j + ((2 5) (4 1))k a b = 23i + 7j + 6k (You have found the cross product) Finding the angle etween 2 vectors: The cross product can also be used to find the angle between two vectors using the following equation: a b = a b sinθ So far we have found that a b = 23i + 7j + 6k Use this to find the modulus (to complete the left hand side): a b = ( 23) 2 + 7 2 + 6 2 = 614 (24. 8) Right hand side: a b = 2 2 + 4 2 + 3 2 1 2 + 5 2 + ( 2) 2 = 29 30 Now we can find the angle between a and b using the cross product equation: a b = a b sinθ 614 = 29 30 sinθ θ = sin 1 ( 614 29 30 ) θ = 57.1 o Parallel or perpendicular? If a b = 0 then sinθ = 0 θ = 0 o parallel If a b = ab then sinθ = 1 θ = 90 o perpendicular H Jackson 2014/15/16 / Academic Skills 4
Right Hand Rule: The answer to the cross product (a b) is a vector that acts perpendicular to both of the original vectors (a and b) and has magnitude a b sinθ. Useful information: For RH rule: b a = (a b) a a = 0 b b = 0 i i = j j = k k = 0 i j = k j i = k j k = i k j = i k i = j i k = j You could remember the RH rule as follows: If the 2 numbers you are multiplying are: In alphabetical order (going round in a circle: i, j, k, i, j, k etc), answer is: Not in alphabetical order, answer is: -other letter. other letter. The reason it is called the Right Hand Rule isn t covered here but you can always search for it if you re interested. Another way to find the cross product formula: If you don t like using Matrices the following is a way to remember the formula: Step 1: Write the formula out but without the subscript numbers: a b = (a b a b)i (a b a b)j + (a b a b)k (Remember the alternate signs on the brackets) Now we just need to fill in the subscript numbers with the a s and b s Step 2: Decide on the numbers that go with the a s racket 1, miss out 1, so write 2 then 3 racket 2, miss out 2, so write 1 then 3 racket 3, miss out 3, so write 1 then 2 The numbers with the a s: a b = (a 2 b a 3 b)i (a 1 b a 3 b)j + (a 1 b a 2 b)k Step 3: Decide on the numbers that go with the b s In each bracket just use the opposite numbers from what you used for the a s The numbers with the b s: We have now written the complete formula: H Jackson 2014/15/16 / Academic Skills 5