91Ó°ÊÓ

vis parallel to w

$\text{Consider the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{.}$

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

$w=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}$

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

$\text{The vector}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{are parallel if and only if they are scalar}$

$\text{multiple of each other.}$

$\text{Thus, the component vectors of both}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{should give equalratio}$

$\text{Therefore, the condition that shows the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{is paralleltoeachother}$

$\frac{a}{\mathrm{Î\pm }}=\frac{b}{\mathrm{β}}=\frac{c}{\mathrm{γ}}$

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

$\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{is}\frac{a}{\mathrm{Î\pm }}=\frac{b}{\mathrm{β}}=\frac{c}{\mathrm{γ}}\text{.}$

vis perpendicular tow

$\text{Consider the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{.}$

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

$\text{The vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\text{and}\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{are perpendicular if the dot product is zero.}$

$\mathbf{V}·\mathbf{w}=0$

$(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})·(\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k})=0$

$a\mathrm{Î\pm }+b\mathrm{β}+c\mathrm{γ}=0$

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

$\mathbf{w}=\mathrm{Î\pm }\mathbf{i}+\mathrm{β}\mathbf{j}+\mathrm{γ}\mathbf{k}\text{is}a\mathrm{Î\pm }+b\mathrm{β}+c\mathrm{γ}=0$