The direction of a vector. Let us find the angle between two vectors and .Consider the inner product
.
We have
. (*)
Writing the product and absolute values in coordinates, we obtain
. (**)
Example 3. Find an angle between vectors and . By using formula (**), we find
,
Let us determine a condition for vectors to be perpendicular. Suppose that vectors and are perpendicular, i.e., ; then , and
. (7)
This is the condition for vectors to be perpendicular.
z
0 y
x
Consider the angles between a vector and the unit vectors . We denote these angles by
; ; .
Take the product of and any unit vector, say, =
.
By formula (*), the cosine of the angle a from it is
.
Similarly the cosines of the other angles are
, , . (8)
These cosines are called the directional cosines of the vector .
The sum of the squared directional cosines equals one:
.
To prove this, it sufficies to square the cosines by formula (8) and sum them:
.
Example 5. For what a are the vectors
and
perpendicular?
We use the perpendicularity condition (7) and write the inner product of the given vectors in coordinates:
; , a=10.
Date: 20150102; view: 178
