Question

1. compute \\(\vec{u} \times \vec{v}\\) for the following: (a) \\(\vec{u} = \langle 3, 0, 0 \

angle\\), \\(\vec{v} = \langle 0, 5, 0 \
angle\\). (b) \\(\vec{u} = \langle 3, 0, 0 \
angle\\), \\(\vec{v} = \langle 0, 0, 5 \
angle\\). (c) \\(\vec{u} = \langle 3, 0, 0 \
angle\\), \\(\vec{v} = \langle 5, 0, 0 \
angle\\). (d) \\(\vec{u} = \langle 2, 4, 6 \
angle\\), \\(\vec{v} = \langle 1, 2, 3 \
angle\\). (e) \\(\vec{u} = \langle 1, 2, 3 \
angle\\), \\(\vec{v} = \langle 4, -1, -2 \
angle\\). (f) \\(\vec{u} = \langle 1, 1, 0 \
angle\\), \\(\vec{v} = \langle 0, -1, 2 \
angle\\). (g) \\(\vec{u} = \langle 2, 4, 7 \
angle\\), \\(\vec{v} = \langle -1, -3, -5 \
angle\\).

Explanation:

Part (a)

Step1: Recall cross product formula

For \(\vec{u}=\langle u_1, u_2, u_3
angle\) and \(\vec{v}=\langle v_1, v_2, v_3
angle\), \(\vec{u}\times\vec{v}=

$$\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\u_1&u_2&u_3\\v_1&v_2&v_3\end{vmatrix}$$

=\vec{i}(u_2v_3 - u_3v_2)-\vec{j}(u_1v_3 - u_3v_1)+\vec{k}(u_1v_2 - u_2v_1)\)
Here, \(\vec{u}=\langle 3,0,0
angle\), \(\vec{v}=\langle 0,5,0
angle\)

Step2: Substitute values into formula

\(\vec{u}\times\vec{v}=\vec{i}(0\times0 - 0\times5)-\vec{j}(3\times0 - 0\times0)+\vec{k}(3\times5 - 0\times0)\)
\(=\vec{i}(0 - 0)-\vec{j}(0 - 0)+\vec{k}(15 - 0)\)
\(= 0\vec{i}- 0\vec{j}+15\vec{k}=\langle0,0,15
angle\)

Step1: Use cross product formula

\(\vec{u}=\langle 3,0,0
angle\), \(\vec{v}=\langle 0,0,5
angle\)
\(\vec{u}\times\vec{v}=

$$\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\3&0&0\\0&0&5\end{vmatrix}$$

=\vec{i}(0\times5 - 0\times0)-\vec{j}(3\times5 - 0\times0)+\vec{k}(3\times0 - 0\times0)\)

Step2: Calculate each component

\(=\vec{i}(0 - 0)-\vec{j}(15 - 0)+\vec{k}(0 - 0)\)
\(= 0\vec{i}-15\vec{j}+0\vec{k}=\langle0, - 15,0
angle\)

Step1: Apply cross product formula

\(\vec{u}=\langle 3,0,0
angle\), \(\vec{v}=\langle 5,0,0
angle\)
\(\vec{u}\times\vec{v}=

$$\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\3&0&0\\5&0&0\end{vmatrix}$$

=\vec{i}(0\times0 - 0\times0)-\vec{j}(3\times0 - 0\times5)+\vec{k}(3\times0 - 0\times5)\)

Step2: Compute components

\(=\vec{i}(0 - 0)-\vec{j}(0 - 0)+\vec{k}(0 - 0)\)
\(= 0\vec{i}- 0\vec{j}+0\vec{k}=\langle0,0,0
angle\) (Since the vectors are parallel)

Answer:

\(\langle0,0,15
angle\)

Part (b)