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*\mathbf w\|=\|\mathbf v\|\|\mathbf w\|\sin\theta`


`\mathbf v*\mathbf w=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\1&1&0\\1&1&1\end{vmatrix}`

`\mathbf v*\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*\mathbf w=\mathbf i-\mathbf j`

`\implies\|\mathbf v*\mathbf w\|=√(1^2+(-1)^2)=\sqrt2`


`\mathbf v=√(1^2+1^2)=\sqrt2`

`\mathbf w=√(1^2+1^2+1^2)=\sqrt3`


`\sqrt 2=\sqrt2\cdot\sqrt3\sin\theta`

`\frac1{\sqrt3}=\sin\theta`