Question 9 & 10, Exercise 3.3

Solutions of Question 9 & 10 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

A force $\vec{k}-2 \hat{i}+3 \hat{j}+\hat{k}$ acts through the displacement $\vec{S}=2 \hat{i}+\hat{j}-\hat{k}$. Find work done.

We know that \begin{align}W &=\vec{F} \cdot s \\ \Rightarrow W &=(2 \hat{i}+3 \hat{j}+\hat{k}) \cdot(2 \hat{i}+\hat{j}-\hat{k}) \\ \Rightarrow W &=2(2) \div 3(1)+1(-1) \\ \Rightarrow W &=4+3 \quad 1=6 \text { units.}\end{align}

Find the work done by the force $\vec{F}=2 \hat{i}+3 \hat{j}+\hat{k}$ in displacement of an object from a point $A(-2,1,2)$ to the poim $B(5.0,3)$.

Position vector of $A$ is: $\overrightarrow{O A}=2 \hat{i}+\hat{j}+2 \hat{k}$.

The position vector of $B$ is: $$\overrightarrow{O B}=5 \hat{i}+0 \hat{j}+3 \hat{k} \text {. }$$ Displacement vector from $A$ to $B$ \begin{align}A \vec{B}&=O \vec{B}-O \hat{A} \\ &=5 \hat{i}+0 \hat{j}+3 \hat{k}-(-2 \hat{i}+\hat{j}+2 \hat{k}) \\ \Rightarrow \overrightarrow{A B}&=7 \hat{i}-\hat{j}+\hat{k} .\end{align} Now work done is $W=\vec{F} \cdot \overrightarrow{A B}$ $$\Rightarrow W=(2 \hat{i}+3 \hat{j}+\hat{k}) \cdot(7 \hat{i}-\hat{j}+\hat{k})$$ \begin{align}\rightarrow W&=2.7+3 .-1+1.1 \\ \therefore W&=14-3+1=12 \text { units. }\end{align}