Let $\mathbf { v } = 9 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$. Select the correct answer.
A) $| \mathbf { u } \times \mathbf { v } | = 159$
B) $| \mathbf { u } \times \mathbf { v } | = 5$
C) $| \mathbf { u } \times \mathbf { v } | = 1$
D) $| \mathbf { u } \times \mathbf { v } | = 45$
E) $| \mathbf { u } \times \mathbf { v } | = 9$

Question

Quizplus · Accepted Answer

Since the maximum value of $|\mathbf{u} 	imes \mathbf{v}|$ occurs when $\mathbf{u}$ is perpendicular to $\mathbf{v}$, we want $\mathbf{u}$ to have the maximum value of $\frac{|\mathbf{v}|}{|\mathbf{u}|}= \frac{9}{5}$. Therefore, $|\mathbf{u}| = \frac{5}{9}|\mathbf{v}| = \frac{5}{9}\cdot 9 = 5$, so the maximum value of $|\mathbf{u} 	imes \mathbf{v}|$ is $|\mathbf{u} 	imes \mathbf{v}| = |\mathbf{v}|\cdot|\mathbf{u}|\cdot \sin 	heta = 9\cdot 5\cdot 1 = 45$. Thus, the answer is $\boxed{	extbf{(D)}}$.

Let v=9j\mathbf { v } = 9 \mathbf { j }v=9j

Let $\mathbf { v } = 9 \mathbf { j }$