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Chapter 16: Problem 1489

If vector \(\underline{r}\) forms an angle \(\alpha, \beta, \gamma\) with \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) -axis then \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=\) (A) 1 (B) 2 (C) \(-1\) (D) \(-2\)

Short Answer

Expert verified
\( \sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma = 2 \)

Step by step solution

01

Use the identity relating cosines and sines of angles

Using the Pythagorean identity, we know that $$\sin^2 \theta + \cos^2 \theta = 1$$ for any angle \(\theta\). We apply this to each of the angles \(\alpha, \beta, \gamma\) to find expressions for \(\sin^2 \alpha, \sin^2 \beta, \sin^2 \gamma\) in terms of the respective cosines: $$\sin^2 \alpha = 1 - \cos^2 \alpha \qquad (1)$$ $$\sin^2 \beta = 1 - \cos^2 \beta \qquad (2)$$ $$\sin^2 \gamma = 1 - \cos^2 \gamma \qquad (3)$$
02

Add the sine-squared expressions

Next, we add equations (1), (2), and (3) to get an expression for \(\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma\) : \(\) $$\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma = (1 - \cos^2 \alpha) + (1 - \cos^2 \beta) + (1 - \cos^2 \gamma)$$ Simplify the expression: $$\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma = 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma)$$
03

Substitute the sum of cosine squares with 1

Recall that \(\cos^2 \alpha+\cos^2 \beta+\cos^2 \gamma=1\) for the directional cosines. Substitute this into the equation obtained in Step 2: $$ \sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma = 3 - 1 $$
04

Simplify and find the answer

Simplify the expression to get the final answer: $$\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma = 2$$ Thus, the value of \(\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma\) is 2. So the correct answer is (B) 2.

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Most popular questions from this chapter

The Image of point \((1,3,4)\) with respect to the plane \(2 x-y+z+3=0\) is (A) \((3,5,2)\) (B) \((-3,-5,2)\) (C) \((-3,-5,-2)\) (D) \((-3,5,2)\)

\([(2-3 \mathrm{x}) / 6]=[(\mathrm{y}+1) / 2]=[(1-z) /(-2)]\) direction of line \(\overline{(A)}+2,2,2\) (B) \(-1,1,1\) (C) \(-3,2,2\) (D) \(6,2,-2\)

The locus of point of the plane passing through \((\alpha, \beta, \gamma)\) and intersect the axis in \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and the plane which is parallel to such plane is (A) \((\mathrm{x} / \alpha)+(\mathrm{y} / \beta)+(\mathrm{z} / \gamma)=1\) (B) \((\alpha / x)+(\beta / y)+(\gamma / z)=1\) (C) \(x+y+z=1\) (D) \(\mathrm{x}+\mathrm{y}+\mathrm{z}=\alpha \beta \gamma\)

\([(x-1) / 3]=[(y+1) / 2]=[(z-1) / 5]\) and \([(\mathrm{x}-2) / 4]=[(\mathrm{y}-1) / 3]=[(\mathrm{z}+1) /(-2)]\) lines are (A) parallel (B) coincident (C) Intersecting (D) skew

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