Consider the vectors $u=5i+4j+3k$ , $v=−4i−j−k$ and $ w=4i+4j−5k$ . Compute $|5u-4v+w|$

Question

I asked this question yesterday, but could not figure out this particular problem, $|5u-4v+w|$. I have $5\sqrt{5^2+4^2+3^2}-4\sqrt{(-4)^2+(-1)^2+(-1)^2}+\sqrt{4^2+4^2+(-5)^2}$ to get $5\sqrt{50}-4\sqrt{18}+\sqrt{57}$

Answer 1

You are making an error when you say $$\|5u-4v+w\|=5\|u\|-4\|v\|+\|w\|$$ Note In general, $\|x+y\| \neq \|x\| + \|y\|$.

There are couple of ways you can proceed:

1. Either you can first compute the vector $5u-4v+w$ using basic vector algebra and then compute it's magnitude. OR
2. You can use the following \begin{align*} \|a+b+c\|^2 & =(a+b+c) \cdot (a+b+c)\\ &=a \cdot a+ b \cdot b + c \cdot c +2(a \cdot b + b \cdot c + c \cdot a)\\ &=\|a\|^2+\|b\|^2+\|c\|^2+2(a \cdot b + b \cdot c + c \cdot a) \end{align*} Hopefully you can take it from here.