Find the direction angles of the vector u given below. Round your answer to one decimal place. $$\mathbf { u } = ( 7,1 , - 3 \}$$
A) $24.3 ^ { \circ } , 82.5 ^ { \circ } , 113.0 ^ { \circ }$
B) $24.3 ^ { \circ } , 82.5 ^ { \circ } , 23.0 ^ { \circ }$
C) $113.0 ^ { \circ } , 71.5 ^ { \circ } , 24.3 ^ { \circ }$
D) $24.3 ^ { \circ } , 113.0 ^ { \circ } , 82.5 ^ { \circ }$
E) $0.4 ^ { \circ } , 90.5 ^ { \circ } , 113.0 ^ { \circ }$

Question

Quizplus · Accepted Answer

The direction angles of a vector $\mathbf{u} = (a, b, c)$ in 3D space can be found using the formulas $\cos^{-1}\left(\frac{a}{\|\mathbf{u}\|}\right)$, $\cos^{-1}\left(\frac{b}{\|\mathbf{u}\|}\right)$, and $\cos^{-1}\left(\frac{c}{\|\mathbf{u}\|}\right)$ for the angles with the x, y, and z axes respectively. Here, $\|\mathbf{u}\|$ is the magnitude of the vector, calculated as $\sqrt{a^2 + b^2 + c^2}$. For $\mathbf{u} = (7, 1, -3)$, the magnitude is $\sqrt{7^2 + 1^2 + (-3)^2} = \sqrt{49 + 1 + 9} = \sqrt{59}$. Thus, the direction angles are $\cos^{-1}\left(\frac{7}{\sqrt{59}}\right) \approx 24.3^\circ$, $\cos^{-1}\left(\frac{1}{\sqrt{59}}\right) \approx 82.5^\circ$, and $\cos^{-1}\left(\frac{-3}{\sqrt{59}}\right) \approx 113.0^\circ$.

Find the Direction Angles of the Vector U Given Below u=(7,1,−3}\mathbf { u } = ( 7,1 , - 3 \}u=(7,1,−3}

Find the Direction Angles of the Vector U Given Below $\mathbf { u } = ( 7,1 , - 3 \}$