Question

Use the cross product to find the sin(θ) where θ is the angle between v and w where v=i+j and w=i+j+k.

Answer 1

$\|\mathbf v\times\mathbf w\|=\|\mathbf v\|\|\mathbf w\|\sin\theta$

$\mathbf v\times\mathbf w=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\1&1&0\\1&1&1\end{vmatrix}$
$\mathbf v\times\mathbf w=\begin{vmatrix}1&0\\1&1\end{vmatrix}\mathbf i-\begin{vmatrix}1&0\\1&1\end{vmatrix}\mathbf j+\begin{vmatrix}1&1\\1&1\end{vmatrix}\mathbf k$

$\mathbf v\times\mathbf w=\mathbf i-\mathbf j$
$\implies\|\mathbf v\times\mathbf w\|=\sqrt{1^2+(-1)^2}=\sqrt2$

$\mathbf v=\sqrt{1^2+1^2}=\sqrt2$
$\mathbf w=\sqrt{1^2+1^2+1^2}=\sqrt3$

$\sqrt 2=\sqrt2\cdot\sqrt3\sin\theta$
$\dfrac1{\sqrt3}=\sin\theta$