Given $v=2i+j-k$ and $w=i+3k$ and $u$ is a unit vector then find the max value of $[uvw]$
I evaluated the cross product of the given 2 vectors. Then I don't actually know what to do; I'm confused. Like do we need to consider a function which we can maximize but the variables will be more and we cannot involve parameters I think. So what should we do? I'll give the options too.
A) $\sqrt{59}$
B) $\sqrt{60}$
C) $\sqrt{10}+\sqrt{6}$
D) none of these
The answer is a) $\sqrt{59}$. You can see this by considering what the unit vector v needs to be for $v \cdot (a \times b)$ to be maximum (where a and b are the given vectors). This is obviously when v is the unit vector in the direction of $a \times b$.
$a \times b$ is $3i - 7j -k$ and so the maximum of $v \cdot (3i - 7j -k )$ is when $v = \frac{ 3i - 7j -k}{\sqrt{3^2+7^2+1^2}}$.
This gives you the answer.