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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 4 (page 621)

The sum of the squares of the direction cosines of a vector in space add up to _______.

Short Answer

Expert verified

The sum of the squares of the direction cosines of a vector in space add up to $1$ .

Step by step solution

01

Step 1. Given information

We have to fill in the blank

The sum of the squares of the direction cosines of a vector in space add up to _______.

02

Step 2. Find the sum of the squares of the direction cosines of a vector in space.

Let $v = a i + b j + c k$

$伪, 尾, 纬$ are the directional angles of this vector.

And $\cos 伪, \cos 尾, \cos 纬$ are the directional cosines.

role="math" localid="1646807102733" $\cos 伪 = \frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}} \cos 尾 = \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}} \cos 纬 = \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}}$

Now,

$\cos^{2} 伪 + \cos^{2} 尾 + \cos^{2} 纬 = \frac{a^{2}}{a^{2} + b^{2} + c^{2}} + \frac{b^{2}}{a^{2} + b^{2} + c^{2}} + \frac{c^{2}}{a^{2} + b^{2} + c^{2}} = \frac{a^{2} + b^{2} + c^{2}}{a^{2} + b^{2} + c^{2}} = 1$

Therefore, the sum of the squares of the direction cosines of a vector in space add up to $1$ .

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