Q .7. - Calculus

Question

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 $\mathbf{w}$.

Step-by-Step Solution

Answer

Part a)The condition is $\frac{a}{\alpha }=\frac{b}{\beta }=\frac{c}{\gamma }$

Part b) The condition is $a\alpha +b\beta +c\gamma =0$

Step 1 Part a):Given information

v is parallel to w

Step 2 Part a):Simplification

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

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

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

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

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

$\text{ multiple of each other. }$

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

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

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

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

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

Step 3:Given information Part b)

v is perpendicular to w

Step 4:Part b) Simplification

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

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

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

$\mathbf{V}·\mathbf{w}=0$

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

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

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

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