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# 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.

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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: 2015-01-02; view: 847

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