Choose the one alternative that best completes the statement or answers the question.
Find the direction angles of the vector. Round to the nearest degree, if necessary.
-$$\mathbf { v } = 3 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k }$$
A) $\alpha = 86 ^ { \circ } , \beta = 88 ^ { \circ } , \gamma = 97 ^ { \circ }$
B) $\alpha = 62 ^ { \circ } , \beta = 72 ^ { \circ } , \gamma = 203 ^ { \circ }$
C) $\alpha = 68 ^ { \circ } , \beta = 76 ^ { \circ } , \gamma = 139 ^ { \circ }$
D) $\alpha = 65 ^ { \circ } , \beta = 73 ^ { \circ } , \gamma = 149 ^ { \circ }$

Question

Quizplus · Accepted Answer

The direction angles $\alpha$, $\beta$, and $\gamma$ are calculated using the formulas: $\cos \alpha = \frac{a}{\|\mathbf{v}\|}$, $\cos \beta = \frac{b}{\|\mathbf{v}\|}$, $\cos \gamma = \frac{c}{\|\mathbf{v}\|}$, where $\|\mathbf{v}\|$ is the magnitude of the vector. For $\mathbf{v} = 3\mathbf{i} + 2\mathbf{j} - 6\mathbf{k}$, $\|\mathbf{v}\| = \sqrt{3^2 + 2^2 + (-6)^2} = 7$. Thus, $\cos \alpha = \frac{3}{7}$, $\cos \beta = \frac{2}{7}$, $\cos \gamma = \frac{-6}{7}$. Calculating these gives $\alpha \approx 65^\circ$, $\beta \approx 73^\circ$, $\gamma \approx 149^\circ$.

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