Let \( \vec{a},\vec{b},\vec{c} \) are three vectors such that \( |\vec{a}|=2,|\vec{b}|=1,|\vec{c}|=3 \) . If projection of \( \vec{a} \) on \( \vec{b} \) is equal to projection of \( \vec{c} \) on \( \vec{b} \) and \( \vec{a}\perp\vec{c} \) , then value of \( |2\vec{a}+3\vec{b}-2\vec{c}| \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \sqrt{61} \)
The projection of \( \vec{a} \) on \( \vec{b} \) is \( \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \). Equal projections on the same \( \vec{b} \) give
\( \displaystyle \vec{a}\cdot\vec{b}=\vec{c}\cdot\vec{b} \)
and \( \vec{a}\perp\vec{c} \) gives \( \vec{a}\cdot\vec{c}=0 \).
Squaring the magnitude and expanding:
\( \displaystyle |2\vec{a}+3\vec{b}-2\vec{c}|^2=4|\vec{a}|^2+9|\vec{b}|^2+4|\vec{c}|^2+12\,\vec{a}\cdot\vec{b}-12\,\vec{b}\cdot\vec{c}-8\,\vec{a}\cdot\vec{c} \)
\( \displaystyle =4(4)+9(1)+4(9)+12(\vec{a}\cdot\vec{b}-\vec{b}\cdot\vec{c})-8(0) \)
\( \displaystyle =16+9+36+0=61 \)
\( \displaystyle |2\vec{a}+3\vec{b}-2\vec{c}|=\sqrt{61} \)