Question 1.

Find a unit vector that is orthogonal to both $\displaystyle i^\rightharpoonup + j^\rightharpoonup$ and $\displaystyle i^\rightharpoonup + k^\rightharpoonup$

My attempt:

$\displaystyle a^\rightharpoonup = <1, 1, 0> $

$\displaystyle b^\rightharpoonup = <1, 0, 1>$

$\displaystyle a^\rightharpoonup * x^\rightharpoonup = 0$

$\displaystyle <1, 1, 0> * <x_1, x_2, x_3> = 0$

$\displaystyle x_1 + x_2 = 0$

$\displaystyle b^\rightharpoonup * x^\rightharpoonup = 0$

$\displaystyle <1, 0, 1> * <x_1, x_2, x_3> = 0$

$\displaystyle x_1 + x_3 = 0$

Now I have 3 variables and 2 equations. What do I do?

Question 2.

If $\displaystyle a^\rightharpoonup = <3, 0, -1>$, find a vector $\displaystyle b^\rightharpoonup $ such that $\displaystyle comp_{a^\rightharpoonup}b^\rightharpoonup = 2$

My attempt:

$\displaystyle |a^\rightharpoonup| = \sqrt{3^2 + 0^2 + (-1)^2} = \sqrt{10}$

$\displaystyle |b^\rightharpoonup|cos\theta = \frac{a^\rightharpoonup * b^\rightharpoonup}{|a^\rightharpoonup|} = 2$

$\displaystyle 2 = \frac{<3, 0, -1> * <b_1, b_2, b_3>}{\sqrt{10}}$

$\displaystyle 2 = \frac{3b_1 + (-1)b_3}{\sqrt{10}}$

Once again, I havmore variables than the # of equations. What should I doo?

Thanks in advance!