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Question

# Find $\mathbf{v} \times \mathbf{w}$ for the vectors given.$\mathbf{v}=\mathbf{k} ; \mathbf{w}=\mathbf{k}$

Solution

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The given vectors are $v =k;w =k$. The cross product is

$\begin{gather*} v \times w&=&\left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\0 & 0 & 1\\0 & 0 & 1\end{array}\right| \\&=& \left|\begin{array}{cc}0 & 1\\0 & 1\end{array}\right| \mathbf{\vec{i}} - \left|\begin{array}{cc}0 & 1\\0 & 1\end{array}\right| \mathbf{\vec{j}} + \left|\begin{array}{cc}0 & 0\\0 & 0\end{array}\right| \mathbf{\vec{k}} \\&=& \left(\left(0\right)\cdot \left(1\right) - \left(1\right)\cdot \left(0\right)\right) \mathbf{\vec{i}} - \left(\left(0\right)\cdot \left(1\right) - \left(1\right)\cdot \left(0\right)\right) \mathbf{\vec{j}} + \left(\left(0\right)\cdot \left(0\right) - \left(0\right)\cdot \left(0\right)\right) \mathbf{\vec{k}} \\&=& 0 + 0 + 0=0 \end{gather*}$

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