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Chapter 10: Q .50. (page 812)

$\cos β = \frac{1}{4}, \cos γ = \frac{1}{3} .$

Short Answer

Expert verified

$The value of the third direction cosine is \pm \frac{\sqrt{119}}{12}$

Step by step solution

01

Step 1:Given information

$\cos β = \frac{1}{4}, \cos γ = \frac{1}{3} .$

02

Step 2:Simplification

$Consider the direction cosines \cos β = \frac{1}{4}, \cos γ = \frac{1}{3} .$

$The objective is to find the third direction cosine using the result Consider the direction cosines$

$\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$Also, \cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$Now, since \cos β = \frac{1}{4}, \cos γ = \frac{1}{3} .$

$Substitute the values in (1)$

$\cos^{2} α + {(\frac{1}{4})}^{2} + {(\frac{1}{3})}^{2} = 1$

$\cos^{2} α + \frac{1}{16} + \frac{1}{9} = 1$

$\cos^{2} α + \frac{9 + 16}{144} = 1$

$\cos^{2} α + \frac{25}{144} = 1$

$\cos^{2} α = 1 - \frac{25}{144}$

$\cos α = \pm \frac{\sqrt{119}}{12}$

$Hence, the value of the third direction cosine is \pm \frac{\sqrt{119}}{12}$

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Most popular questions from this chapter

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k = 1$ .

(b) True or False: $\sum_{k = 0}^{n} \frac{1}{k + 1} + \sum_{k = 1}^{n} k^{2}$ is equal to $\sum_{k = 0}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(c) True or False: $\sum_{k = 1}^{n} \frac{1}{k + 1} + \sum_{k = 0}^{n} k^{2}$ is equal to $\sum_{k = 1}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(d) True or False: $(\sum_{k = 1}^{n} \frac{1}{k + 1}) (\sum_{k = 1}^{n} k^{2})$ is equal to $\sum_{k = 1}^{n} \frac{k^{2}}{k + 1}$ .

(e) True or False: $\sum_{k = 0}^{m} \sqrt{k} + \sum_{k = m}^{n} \sqrt{k}$ is equal to $\sum_{k = 0}^{n} \sqrt{k}$ .

(f) True or False: $\sum_{k = 0}^{n} a_{k} = - a_{0} - a_{n} + \sum_{k = 1}^{n - 1} a_{k}$ .

(g) True or False: ${(\sum_{k = 1}^{10} a_{k})}^{2} = \sum_{k = 1}^{10} a_{k}^{2}$ .

(h) True or False: $\sum_{k = 1}^{n} {(e^{x})}^{2} = \frac{e^{x} (e^{x} + 1) (2 e^{x} + 1)}{6}$ .

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.

$u = <- 5, 1, 3>, v = <- 3, 2, 7>$

Consider the function f shown in the graph next at the right. Use the graph to make a rough estimate of the average value of f on [−4, 4], and illustrate this average value as a height on the graph.

Find a unit vector in the direction opposite to $<3, 4, 5>$ .

Use calculator graphs to make approximations for each of the limits in Exercises 67–74.

$\lim_{x \to 4} (3 - 4 x - 5 x^{2})$

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