Let $\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$and $\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{j}$. Give conditions on the constants $a,b,c,\alpha ,\beta$, and $\gamma$that guarantee that

(a) $\mathbf{v}$is parallel to $\mathit{w}$.

(b) $\mathbf{v}$ is perpendicular to $\mathbf{w}$.

vis parallel to w

$\text{Consider the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{.}$

$\text{The objective is to give conditions on the constants that guarantee}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{is parallel to}$

$w=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}$

The vectors are parallel to each other is they are scalar multiple of each other.

$\text{The vector}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{are parallel if and only if they are scalar}$

$\text{multiple of each other.}$

$\text{Thus, the component vectors of both}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{should give equalratio}$

$\text{Therefore, the condition that shows the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{is paralleltoeachother}$

$\frac{a}{\alpha }=\frac{b}{\beta }=\frac{c}{\gamma }$

$\text{The condition conditions on the constants that guarantee}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{is parallel to}$

$\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{is}\frac{a}{\alpha }=\frac{b}{\beta }=\frac{c}{\gamma }\text{.}$

vis perpendicular tow

$\text{Consider the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{.}$

$\text{The vectors are perpendicular to each other is their dot product is zero.}$

$\text{The vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{are perpendicular if the dot product is zero.}$

$\mathbf{V}·\mathbf{w}=0$

$(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})·(\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k})=0$

$a\alpha +b\beta +c\gamma =0$

$\text{Therefore, the condition on the constants that guarantee}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{is perpendicular to}$

$\mathbf{w}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}\text{is}a\alpha +b\beta +c\gamma =0$