The direction of a vector. Let us find the angle between two vectors and .
Consider the inner product
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
This is the condition for vectors to be perpendicular.
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
We use the perpendicularity condition (7) and write the inner product of the given vectors in coordinates:
; , a=10.
Date: 2015-01-02; view: 308