If \( \vec{a},\vec{b} \) and \( \vec{c} \) are mutually perpendicular vectors and \( |\vec{a}|=2,\ |\vec{b}|=3 \) and \( |\vec{c}|=4 \), then the value of \( \left|3\vec{a}-2\vec{b}+\vec{c}\right| \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 2\sqrt{22} \)
The vectors are mutually perpendicular, so \( \vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{c}=\vec{c}\cdot\vec{a}=0 \), and the mixed terms vanish when the square of the magnitude is expanded:
\( \displaystyle |3\vec{a}-2\vec{b}+\vec{c}|^{2} \)
\( \displaystyle =9|\vec{a}|^{2}+4|\vec{b}|^{2}+|\vec{c}|^{2} \)
\( \displaystyle =36+36+16=88 \)
Hence
\( \displaystyle |3\vec{a}-2\vec{b}+\vec{c}|=\sqrt{88}=2\sqrt{22} \)