Originally Posted by

**chug1** Well the length of a vector [a,b,c,...] is

$\displaystyle

\sqrt[]{{a^2}+{b^2}+{c^2}+...}

$

so the legnth of u would be

$\displaystyle \sqrt[]{{1^2}+{2^2}+{3^2}+{4^2}}$

$\displaystyle =\sqrt[]{{1}+{4}+{9}+{16}}$

$\displaystyle =\sqrt[]{30}$

which is what you got, you just need to provide it as a squareroot for the "exact answer"

Sorry i can't help with (b), but for (c) a vector in the opposite direction has the negative of the components of the original vector. e.g. the opposite direction to [a,b,c,d] is [-a,-b,-c,-d], so the opposite direction to w=[0,2,-2,2] is [0,-2,2,-2]. A **unit** vector in this direction has a length of 1, so it is some positive multiple k of [0,-2,2,-2] which has a length of 1. Calculate the length using the above formula.

$\displaystyle 1=k\:\sqrt[]{{0^2}+{(-2)^2}+{2^2}+{(-2)^2}}$

$\displaystyle 1=k\:\sqrt[]{0+4+4+4}$

$\displaystyle 1=k\:\sqrt[]{12}$

$\displaystyle k=\frac{1}{\sqrt[]{12}}$

so the unit vector in the direction opposite to w=[0,2,-2,2] is

$\displaystyle

\frac{1}{\sqrt[]{12}}\:\left[0,-2,2,-2 \right]

$

Hope this helps