The unit vector which is orthogonal to the vector $3 \hat{i}+2 \hat{j}+6 \hat{k}$ and coplanar with the…

The unit vector which is orthogonal to the vector $3 \hat{i}+2 \hat{j}+6 \hat{k}$ and coplanar with the vectors $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$ is
  1. $\frac{8 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}}{\sqrt{82}}$
  2. $\frac{-8 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}}{\sqrt{82}}$
  3. $\frac{-8 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}}{\sqrt{82}}$
  4. $-\frac{8 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}}{\sqrt{82}}$

Solution

Consider option (C) $(3 \hat{i}+2 \hat{j}+6 \hat{k}) \cdot\left(\frac{-8 \hat{i}+3 \hat{j}+3 \hat{k}}{\sqrt{82}}\right)=0$ This is valid for only option (C) $\therefore \quad$ Option (C) is correct.

Asked in: MHT CET 2023 (12 May Shift 2)

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