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

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 _______.

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

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

$α, β, γ$ are the directional angles of this vector.

And $\cos α, \cos β, \cos γ$ are the directional cosines.

role="math" localid="1646807102733" $\cos α = \frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}} \cos β = \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}} \cos γ = \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}}$

Now,

$\cos^{2} α + \cos^{2} β + \cos^{2} γ = \frac{a^{2}}{a^{2} + b^{2} + c^{2}} + \frac{b^{2}}{a^{2} + b^{2} + c^{2}} + \frac{c^{2}}{a^{2} + b^{2} + c^{2}} = \frac{a^{2} + b^{2} + c^{2}}{a^{2} + b^{2} + c^{2}} = 1$

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

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

Short Answer

Step by step solution

Step 1. Given information

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

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