Let $v = 7 j$ and let u be a vector with length $5$ that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .
A) $| \mathbf { u } \times \mathbf { v } | = 5$
B) $| \mathbf { u } \times \mathbf { v } | = 7$
C) $| \mathbf { u } \times \mathbf { v } | = 35$
D) $| \mathbf { u } \times \mathbf { v } | = 157$
E) $| \mathbf { u } \times \mathbf { v } | = 1$

Question

Quizplus · Accepted Answer

Since $|u	imes v|$ is the area of the smallest parallelogram that contains both $u$ and $v$, it makes sense to make $u$ and $v$ orthogonal (so that the parallelogram is a rectangle) in order to maximize the area. The maximum value of $|u	imes v|$ therefore occurs when $u$ is a vector in the $xy$-plane perpendicular to $v$. We can find this vector by rotating $v$ counterclockwise by $90$ degrees: $$u = \begin{pmatrix} 0 & -1 \ 1 & 0\end{pmatrix}v = \begin{pmatrix} 0 \ -7\end{pmatrix}.$$ Thus, $$|u	imes v| = \left|\begin{pmatrix} 0 \ -7 \ 0 \end{pmatrix} 	imes \begin{pmatrix} 7 \ 0 \0 \end{pmatrix} ight| = \left|\begin{pmatrix} 0 \ 0 \ 49\end{pmatrix}ight| = \boxed{35}.$$

Let v=7jv = 7 jv=7j And Let U Be a Vector with Length

Let $v = 7 j$ And Let U Be a Vector with Length