Q .6. - Calculus

Question

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

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

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

Step-by-Step Solution

Answer

Part a)The condition is $\frac{a}{c}=\frac{b}{d}$

Part b)The condition is $ac+bd=0$

Step 1 Part a:Given information

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

 v is parallel to w 

Step 2:Simplification Part a)

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

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

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

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

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

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

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

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

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

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

Step 3 Part b :Given information

 v is perpendicular to w

Step 4:Explaination Part b)

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

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

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

 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: }$

$\mathbf{v}·\mathbf{w}=0$

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

$ac+bd=0$

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

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