Question 9 Exercise 3.5

Solutions of Question 9 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.

Write the value of $(\hat{i} \times \hat{j}). \hat{k}+\hat{i}. \hat{j}$

$$\begin{align} (\hat{i} \times \hat{j}) \cdot \hat{k}&=\left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right|&=1 ....(1)\\ \text { and } \hat{i} \cdot \hat{j}&=0 \quad \because \quad \hat{i} \perp \hat{j}\end{align}$$ Adding the above two, we get

$$(i \times \hat{j}) \cdot \hat{k}+\hat{i} \cdot \hat{j}=1+0=1$$ .

Write the value of $(\hat{k} \times \hat{j}) \cdot \hat{i}+\hat{j} \cdot \hat{k}$

$$\begin{align} (\hat{k} \times \hat{j}) \cdot \hat{i}&=\left|\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right|&=-1 ....(1)\\ \hat{j} \cdot \hat{k}&=0 \quad \because \quad \hat{j} \perp \hat{k}\end{align}$$ Adding the above two, we get

$$(\hat{k} \times \hat{j}) \cdot \hat{i}+\hat{j} \cdot \hat{k}=-1+0=-1 \text {. }$$

< Question 8