Vectors
– Addition, Dot Product, Direction Cosines
Interpretation of Dot Product Let θ be the angle between U and V . U ∙
V = |U|
|V| cos θ
So cos θ
= U ∙ V / |U| |V One can use this dot product to calculate the angle between a vector U and each coordinate axis, x, y, and z. Call them θx, θy, θz . Then the cosine of these angles are
called the “direction cosines” .
i.e. cosθx = U ∙ i / |U| |i| = u1 / |U| , cosθy = U ∙ j / |U| |j| = u2 / |U| cosθz = U ∙ k / |U| |k| = u3 / |U| |
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Example of a Dot Product Note: i ∙
i
= 1 , j
∙ j = 1
, k ∙ k = 1 ,
i ∙
j = 0, i ∙
k = 0, j
∙ k = 0
Let U = 3 i + 4 j + 5 k and V = - i + j - 6 k U ∙ V = (3)(- 1) + (4)(1) + (5)(- 6) = - 31 (scalar result) Click here to continue with discussion on scalar and vector projections. |
Copyright © 2017 Richard C. Coddington
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