Question

1)If P>Q =4i+3j+2k
Find the magnitude of PQ' i.e /PQ/
(2) if A'=3i-j+2k and'B=2i+3j-k, Find
(1) 'A×B'
(2) A'×(2A'+3B')
(3)A'.B'

Answer 1

Assume

$P=4\hat{i}+3\hat{j}+2\hat{k}$

$Q=2\hat{i}+3\hat{j}+\hat{k}$

$|P|=\sqrt{16+9+4}=\sqrt{30}\\|Q|=\sqrt{4+9+1}=\sqrt{14}$

$P>Q$ correct statement

PQ=$(4\hat{i}+3\hat{j}+2\hat{k}).(2\hat{i}+3\hat{j}+\hat{k})$

PQ=8+9+2=19

2.

$A=3\hat{i}-\hat{j}+2\hat{k}\\B=2\hat{i}+3\hat{j}-\hat{k}$

$A\times B=\begin{vmatrix} \hat{i} & \hat{j}&\hat{k} \\ 3& -1&2\\2&3&-1 \end{vmatrix}$

$$A\times B=\hat{i}(1-6)-\hat{j}(-3-4)+\hat{k}(9+2)$$

$A\times B=-5\hat{i}+7\hat{j}+11\hat{k}$

(2)

$$2A+2B=2\times(3\hat{i}-\hat{j}+2\hat{k})+2\times (2\hat{i}+3\hat{j}-\hat{k})$$

$(2A+2B)=10\hat{i}+4\hat{j}+2\hat{k}$

$A\times(2A+2B)=\begin{vmatrix} \hat{i} & \hat{j}&\hat{k} \\ 3& -1&2\\10&4&2 \end{vmatrix}$

$$A\times(2A+2B)=\hat{i}(-2-6)-\hat{j}(6-20)+\hat{k}(12+10)=-8\hat{i}+14\hat{j}+22\hat{k}$$

$$A.B=(3\hat{i}-\hat{j}+2\hat{k}).(2\hat{i}+3\hat{j}-\hat{k})=6-6-2=-2$$