Question 4 Exercise 3.4

Solutions of Question 4 of Exercise 3.4 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.

If $\vec{a}=3 \hat{i}-6 \hat{j}+5 \hat{k},\quad\vec{b}=2\hat{i}-\hat{j}+4 \hat{k} \quad$ and $\quad \vec{c}=\hat{i}+\hat{j} \quad \hat{k},\quad$ find $\vec{a} \times \vec{b}$

\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 3 & -6 & 5\\ 2 & -1 & 4\\ \end{array}\right| \\ & =(-24+5) \hat{i}-(12-10) \hat{j}+(-3+12) \hat{k}\\ \Rightarrow \vec{a} \times \vec{b}&=-19 \hat{i}-2 \hat{j}+9 \hat{k} .\end{align}

If $\vec{a}=3 \hat{i}-6 \hat{j}+5 \hat{k},\quad\vec{b}=2\hat{i}-\hat{j}+4 \hat{k}\quad$ and $\quad\vec{c}=\hat{i}+\hat{j} \quad \hat{k},\quad$ find $\vec{b} \times \vec{c}$

\begin{align}\vec{b} \times \vec{c}&=\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 4 \\ 1 & 1 & -1 \end{array}\right| \\ &=(1-4) \hat{i}-(-2-4) \hat{j}+(2+1) \hat{k} \\ \Rightarrow \vec{b} \times \vec{c}&=-3 \hat{i}+6 \hat{j}+3 \hat{k} .\end{align}

If $\vec{a}=3 \hat{i}-6 \hat{j}+5 \hat{k},\quad\vec{b}=2\hat{i}-\hat{j}+4 \hat{k}\quad$ and $\quad\vec{c}=\hat{i}+\hat{j} \quad \hat{k},\quad$ find $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$

\begin{align}\vec{a}+\vec{b}&=(3 \vec{i}-6 \hat{j}+5 \hat{k})+(2 \hat{i}-\hat{j}+4 \hat{k}) \\ \Rightarrow \quad \vec{a}+\vec{b}&=5 \hat{i}-7 \hat{j}+9 \hat{k} \ldots \ldots \ldots \ldots(1) \\ \vec{a}-\vec{b}&=(3 \hat{i}-6 \hat{j}+5 \hat{k})-(2 \hat{i}-\hat{j}+4 \hat{k})\\ \Rightarrow \vec{a}-\vec{b}&=\hat{i}-5 \hat{j}+\hat{k} \ldots \ldots \ldots . .(2) \\ (\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 5 & -7 & 9 1 & -5 & 1 \end{array}\right| \\ \Rightarrow(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})&=(-7+45) \hat{i}-(5-9) \hat{j}+(-25+7) \hat{k} \\ \Rightarrow(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})&=38 \hat{i}+4 \hat{j}-18 \hat{k} .\end{align}