Find the length and direction (when defined) of u × v.
-u = 6i, v = 7j
A) 6; 7k
B) 42; 42k
C) 42; k
D) 42; -k

Question

Quizplus · Accepted Answer

To find the cross product of two vectors, we first need to compute the determinant of the following matrix:

$$\begin{vmatrix}
\bold{i} & \bold{j} & \bold{k} \
6 & 0 & 0 \
0 & 7 & 0 \
\end{vmatrix}$$

Expanding the determinant with respect to the first column (since i has a coefficient of 6 in the first row), we get:

$$\begin{vmatrix}
0 & 0 \
7 & 0 \
\end{vmatrix} \bold{i}
- \begin{vmatrix}
6 & 0 \
0 & 0 \
\end{vmatrix} \bold{j}
+ \begin{vmatrix}
6 & 0 \
0 & 7 \
\end{vmatrix} \bold{k} = 0 + 0\bold{i} - 0\bold{j} + 42\bold{k}$$

Therefore, the cross product of u and v is 42k. The length of the cross product is |u x v| = |42k| = 42, so the correct answer is (C) 42; k.

Find the Length and Direction (When Defined) of U ×