Maximizing vector scalar product

Suppose we have vectors as $$\vec{V} = 2\hat{i} + \hat{j} -\hat{k}$$ and

$$\vec{W} = \hat{i} + 3\hat{k}$$

So what will maximum value of $$k=[ \vec{U} \vec{V} \vec{W}]$$ for some unit vector $\vec{U}$ ?

I just came up with $$\vec{V} \times\vec{W}$$ which is equal to $3\hat{i} -7 \hat{j} -\hat{k}$ . Now to maximse I took dot product in a way that it all add up by supposing vector $\vec{U} = \frac{\hat{i} - \hat{j} -\hat{k}}{\sqrt{3}}$ but came with wrong answer so how to do it?

1 Answer

After thinking a bit I realized that dot product can be maximum only if both the vectors are same so dot product with $\vec{V} \times \vec{W}$ and its unit vector maximized scalar triple product.