Q73 / 180 · KCET 2015 · Mathematics

If \( \vec{a}=\hat{i}+2 \hat{j}+2 \hat{k},|\vec{b}|=5 \) and the angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\Pi}{6} \), then the area of the
triangle formed by these two vectors as two sides is

  • A
    \( \frac{15}{2} \)
  • B
    \( 15 \)
  • Correct answer
    C
    \( \frac{15}{4} \)
  • D
    \( \frac{15 \sqrt{3}}{2} \)

Solution

Given that $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k} \rightarrow(1)$ So, $|\vec{a}|=\sqrt{1+4+4}=\sqrt{9}=3$ $|\vec{b}|=5$ and angle between $\vec{a}$ and $\vec{b}$ is $\frac{\Pi}{6}$ We know that area formed by two vectors as two sides is given by Area $=\frac{1}{2}|\vec{a}| \vec{b} \mid \sin \theta$ $=\frac{1}{2}(3)(5) \sin \frac{\Pi}{6}=\frac{15}{4}$

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

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