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Chapter 10: Q .11 (page 811)

What geometric relationship must two vectors have in order for鈥u+v鈥=鈥u鈥+鈥v鈥 ?

Short Answer

Expert verified

The geometric relationship between the vectors is that they are parallel to each other and are in same direction.

Step by step solution

01

Step 1:Given information

The given expression is鈥u+v鈥=鈥u鈥+鈥v鈥

02

Step 2:Simplification

Consider the non-zero vectorsuandv.

The objective is to find the geometric relationship between two vectors if鈥u+v鈥=鈥u鈥+鈥v鈥.

To find the relationship, consider the value鈥u+v鈥.

The expression鈥u+v鈥gives:

鈥u+v鈥2=(u+v)路(u+v)

鈥u+v鈥2=鈥u鈥2+鈥v鈥2+2鈥u鈥鈥v鈥cos胃鈥鈥

For胃=0掳, the equation (1) reduces to:

鈥u+v鈥2=鈥u鈥2+鈥v鈥2+2鈥u鈥v鈥

鈥u+v鈥2=(鈥u鈥+鈥v鈥)2

鈥u+v鈥=鈥u鈥+鈥v鈥

The result鈥u+v鈥=鈥u鈥+鈥v鈥holds when胃=0掳.

Therefore, the geometric relationship between the vectors is that they are parallel to each other and are in same direction.

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