Multivariate Calculus - Bruce E. Shapiro

16 LECTURE 2. VECTORS IN 3D
Example 2.5 Find ˆv, where v = (3, 0, −4).
Solution. Since
‖v‖ = √ 3 2 + (−4) 2 = 5
then
ˆv =
v ( ) 3
‖v‖ = 5 , 0, −4
5
Three special unit vectors that are parallel to the x, y and z axes are often
defined,
i = (1, 0, 0) (2.25)
j = (0, 1, 0) (2.26)
k = (0, 0, 1) (2.27)
In terms of i, j and k, any vector v = (v x , v y , v z ) can also be written as
v = (v x , v y , v z )
= (v x , 0, 0) + (0, v y , 0) + (0, 0, v z )
= v x (1, 0, 0) + v y (0, 1, 0) + v z (0, 0, 1)
= v x i + v y j + v z k
The unit vectors i, j and k are sometimes called the basis vectors of Euclidean
space. Since i, j and k are always unit vectors, we will usually refer to them without
the “hat.”
If v = (v x , v y , v z ) then
v · i = (v x , v y , v z ) · (1, 0, 0) = v x
v · j = (v x , v y , v z ) · (0, 1, 0) = v y
v · k = (v x , v y , v z ) · (0, 0, 1) = v z
Definition 2.9 The direction angles {α, β, γ} of a vector are the angles between
the vector and the three coordinate axes.
Definition 2.10 The direction cosines of a vector are the cosines of its direction
angles.
Since the vectors i, j, k are parallel to the three coordinate axes, we have
Thus the three direction cosines are
v · i = ‖v‖ cos α
v · j = ‖v‖ cos β
v · k = ‖v‖ cos γ
cos α = v · i
‖v‖ , cos β = v · j
‖v‖ , cos γ = v · k
‖v‖
(2.28)
Revised December 6, 2006. Math 250, Fall 2006