91Ó°ÊÓ

$\mathbf{v}=a\mathbf{i}+b\mathbf{j}\text{and}\mathbf{w}=c\mathbf{i}+d\mathbf{j}$

v is parallel to w

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

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

$\mathbf{w}=c\mathbf{i}+d\mathbf{j}$

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

$\text{The vector}\mathbf{v}=a\mathbf{i}+b\mathbf{j}\text{and}\mathbf{w}=c\mathbf{i}+d\mathbf{j}\text{are parallel if and only if they are scalar multiple of eachother}$

$\text{Thus, the component vectors of both}\mathbf{v}=a\mathbf{i}+b\mathbf{j}\text{and}\mathbf{w}=c\mathbf{i}+d\mathbf{j}\text{should give equal ratio.}$

$\text{Therefore, the condition that shows the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}\text{and}\mathbf{w}=c\mathbf{i}+d\mathbf{j}\text{is parallel to eachotheris}$

$\frac{a}{c}=\frac{b}{d}$

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

$\frac{a}{c}=\frac{b}{d}$

v is perpendicular tow

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

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

$\mathbf{w}=c\mathbf{i}+d\mathbf{j}$

The vectors are perpendicular to each other is their dot product is zero.

$\text{The vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}\text{and}\mathbf{w}=c\mathbf{i}+d\mathbf{j}\text{are perpendicular if the dot product is zero.}$$\text{The dot product of the vectors}\mathbf{v}=a\mathbf{i}+b\mathbf{j}\text{and}\mathbf{w}=c\mathbf{i}+d\mathbf{j}\text{is:}$

$\mathbf{v}·\mathbf{w}=0$

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

$ac+bd=0$

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

$\mathbf{w}=c\mathbf{i}+d\mathbf{j}\text{is}ac+bd=0$