# Question 1 & 2 Exercise 3.5

Solutions of Question 1 & 2 of Exercise 3.5 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.

Find $\vec{a} \cdot \vec{b} \times \vec{c}$. if $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$, $\vec{b}=-\hat{i}+2 \hat{j}+\hat{k} \quad \text { and }\quad \vec{c}=3 \hat{i}+\hat{j}+2 \hat{k} \text {. }$

We know that \begin{align}V&=\vec{a} \cdot \vec{b} \times \vec{c}\\ &=\left|\begin{array}{ccc} 2 & 1 & 3 \\ -1 & 2 & 1 \\ 3 & 1 & 2 \end{array} \right| \\ \Rightarrow V& =2(4-1)-1(-2-3)+3(-1-6) \\ \Rightarrow V&=6+5-21=-10 \text {. }\text{unit cub}\end{align}

Find the volume of the parallelopiped whose edges are represented by $\vec{a}=3 \hat{i}+\hat{j}-\hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+\hat{k}\quad$ and $\quad\vec{c}=\hat{i}-3 \hat{j}-4 \hat{k}$

The volume of parallelopiped is: \begin{align}V&=\vec{a} \cdot \vec{b} \times \vec{c}\\ &=\left|\begin{array}{ccc} 3 & 1 & -1 \\ 2 & -3 & 1 \\ 1 & -3 & -4 \end{array}\right|\\ \Rightarrow V&=3(12+3)-1(-8-1)-(-6,3) \\ V&=45+9+3=57 \text { unit cube. }\end{align}