1
Question
Let −−→PR=3^i+^j−2^k and −−→SQ=^i−3^j−4^k determine diagonals of a parallelogram PQRS and −−→PT=^i+2^j+3^k be another vector. Then the volume (in cubic units) of the parallelopiped determined by the vectors −−→PT,−−→PQ and −→PS is
Let −−→PR=3^i+^j−2^k and −−→SQ=^i−3^j−4^k determine diagonals of a parallelogram PQRS and −−→PT=^i+2^j+3^k be another vector. Then the volume (in cubic units) of the parallelopiped determined by the vectors −−→PT,−−→PQ and −→PS is
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Solution
The correct option is C 10
−→PS=−−→QR, −−→PQ=−−→SR
∴−−→PR=−−→PQ+−→PS=3^i+^j−2^k ⋯(1)
⇒−−→SQ=−−→PQ−−→PS=^i−3^j−4^k ⋯(2)
From (1) and (2),
−−→PQ=2^i−^j−3^k
and −→PS=^i+2^j+^k
Volume of the parallelopiped with co-terminous vectors −−→PT,−−→PQ and −→PS is −−→PT⋅(−−→PQ×−→PS)
=∣∣
∣∣∣∣
∣∣1232−1−3121∣∣
∣∣∣∣
∣∣=10 cubic units.
−→PS=−−→QR, −−→PQ=−−→SR
∴−−→PR=−−→PQ+−→PS=3^i+^j−2^k ⋯(1)
⇒−−→SQ=−−→PQ−−→PS=^i−3^j−4^k ⋯(2)
From (1) and (2),
−−→PQ=2^i−^j−3^k
and −→PS=^i+2^j+^k
Volume of the parallelopiped with co-terminous vectors −−→PT,−−→PQ and −→PS is −−→PT⋅(−−→PQ×−→PS)
=∣∣ ∣∣∣∣ ∣∣1232−1−3121∣∣ ∣∣∣∣ ∣∣=10 cubic units.
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