Question:
Let \(\left|\vec{A_1}\right| = 3\), \(\left|\vec{A_2}\right| = 5\) and \(\left|\vec{A_1} + \vec{A_2}\right| = 5\). The value of \(\left(2\vec{A_1} + 3\vec{A_2}\right)\cdot\left(3\vec{A_1} - 2\vec{A_2}\right)\) is :-
Let \(\left|\vec{A_1}\right| = 3\), \(\left|\vec{A_2}\right| = 5\) and \(\left|\vec{A_1} + \vec{A_2}\right| = 5\). The value of \(\left(2\vec{A_1} + 3\vec{A_2}\right)\cdot\left(3\vec{A_1} - 2\vec{A_2}\right)\) is :-
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Find \(\vec{A_1}\cdot\vec{A_2}\) from \(\left|\vec{A_1} + \vec{A_2}\right|^2\) and expand the dot product.
Updated On: Nov 5, 2026
- -118.5
- -112.5
- -99.5
- -106.5
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The Correct Option is A
Solution and Explanation
\(\left|\vec{A_1} + \vec{A_2}\right|^2 = 9 + 25 + 2\,\vec{A_1}\cdot\vec{A_2} = 25\), so \(\vec{A_1}\cdot\vec{A_2} = -4.5\).
\(\left(2\vec{A_1} + 3\vec{A_2}\right)\cdot\left(3\vec{A_1} - 2\vec{A_2}\right) = 6|\vec{A_1}|^2 - 4\,\vec{A_1}\cdot\vec{A_2} + 9\,\vec{A_1}\cdot\vec{A_2} - 6|\vec{A_2}|^2\)
\(= 6(9) + 5(-4.5) - 6(25) = 54 - 22.5 - 150 = -118.5\).
Answer: (1) -118.5
\(\left(2\vec{A_1} + 3\vec{A_2}\right)\cdot\left(3\vec{A_1} - 2\vec{A_2}\right) = 6|\vec{A_1}|^2 - 4\,\vec{A_1}\cdot\vec{A_2} + 9\,\vec{A_1}\cdot\vec{A_2} - 6|\vec{A_2}|^2\)
\(= 6(9) + 5(-4.5) - 6(25) = 54 - 22.5 - 150 = -118.5\).
Answer: (1) -118.5
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