Let a = 3i + j - 2k, b = 4i + j + 7k and c = i - 3j + 4k

Question

Let \(\overrightarrow{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \overrightarrow{b}=4 \hat{i}+\hat{j}+7 \hat{k} \) and \(\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) be three vectors. If a vectors \(\overrightarrow{\mathrm{p}}\) satisfies \(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{a}}=0\), then \(\overrightarrow{\mathrm{p}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})\) is equal to

(1) 24

(2) 36

(3) 28

(4) 32

Answer 1

Correct option is (4) 32

\(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0}\)

\((\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}=\overrightarrow{0}\)

\(\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{c}}=\lambda \overrightarrow{\mathrm{b}} \Rightarrow \overrightarrow{\mathrm{p}}=\overrightarrow{\mathrm{c}}+\lambda \overrightarrow{\mathrm{b}}\)

Now, \(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{a}}=0\) (given)

So, \(\overrightarrow{c} . \overrightarrow{a}+\lambda\overrightarrow{a} \cdot\overrightarrow{b}=0\)

\((3-3-8)+\lambda(12+1-14)=0\)

\(\lambda=-8\)

\(\overrightarrow{\mathrm{p}}=\overrightarrow{\mathrm{c}}-8 \overrightarrow{\mathrm{b}}\)

\(\overrightarrow{p}=-31 \hat{i}-11 \hat{j}-52 \hat{k}\)

So, \(\overrightarrow{\mathrm{p}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})\)

\(=-31+11+52\)

\(=32\)