Find conditions on vectors

Question

I'm a bit stuck on this question. Please help.

Find conditions on $a$ and $b$ so that the two vectors $ae_1 −2e_2 + 4e_3$ and $e_1 −be_2 −e_3$ are

1. perpendicular;
2. parallel.

Thanks.

Answer 1

Some definitions:

$$\vec v\ \bot\ \vec w \iff \vec v \cdot \vec w = 0 \\ \vec v\ \|\ \vec w \iff \vec v =\lambda \vec w\ \vee \lambda\vec v =\vec w,\quad \lambda\in \Bbb R$$

In case you can't understand that, I'll write those two definitions in words.
"A vector $\vec v$ is orthogonal (perpendicular) to the vector $\vec w$ if and only if the dot product of $\vec v$ and $\vec w$ is $0$."
"A vector $\vec v$ is parallel to the vector $\vec w$ if and only if $\vec v$ is a real multiple of $\vec w$ or $\vec w$ is a real multiple of $\vec v$ (these are not completely equivalent because one might be the zero vector and the other not)."

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So you've got $\vec v= a\hat e_1 -2\hat e_2 +4\hat e_3$ and $\vec w = \hat e_1 -b\hat e_2 - \hat e_3$. Let's apply the definitions.

First orthogonality: $$\vec v \cdot \vec w = 0 \\ \left(a\hat e_1 -2\hat e_2 +4\hat e_3\right) \cdot \left(\hat e_1 -b\hat e_2 - \hat e_3\right)=0 \\ a+2b-4=0$$

Thus $\vec v\ \bot\ \vec w$ if $a$ and $b$ satisfy the condition $a+2b=4$.

Next parallelness:

$$\vec v = \lambda \vec w,\quad \lambda\in\Bbb R \\ a\hat e_1 -2\hat e_2 +4\hat e_3 = \lambda\left(\hat e_1 -b\hat e_2 - \hat e_3\right),\quad \lambda\in\Bbb R \\ a\hat e_1 -2\hat e_2 +4\hat e_3 = \lambda\hat e_1 -b\lambda\hat e_2 - \lambda\hat e_3,\quad \lambda\in\Bbb R \\ \iff \begin{cases} a=\lambda \\ -2=-b\lambda \\ 4=-\lambda\end{cases}\qquad \lambda\in\Bbb R$$

That last equation tells us that $\lambda =-4$ so plugging that into the other two equations we see that $\vec v\ \|\ \vec w$ when $a=-4$ and $b=-\frac 12$.