Q74 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics

If $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and $-\hat{i}+2 \hat{j}+\hat{k}$ are the two diagonals of a parallelogram, then the area of the parallelogram in square units is

  • A
    $\frac{1}{2} \sqrt{170}$
  • B
    $\sqrt{174}$
  • Correct answer
    C
    $\sqrt{\frac{87}{2}}$
  • D
    $\frac{1}{4} \sqrt{174}$

Solution

Let $\overrightarrow{\mathrm{d}}_1=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}}_2=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ Now Area is $=\frac{1}{2} \overline{\mathrm{d}}_1 \times \overline{\mathrm{d}}_2$ $=\frac{1}{2}(2 \hat{i}+3 \hat{j}-4 \hat{k}) \times(-\hat{i}+2 \hat{j}+\hat{k})=\sqrt{\frac{87}{2}}$

Hence, the correct answer is option: 'C'.

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