If P→$$=3i^+5j^+2k^$$ , Q→$$=5i^$$−$$j^+4k^$$ and R→$$=4i^+2j^+3k^$$ are coplanar, then:

Question

If  P→$$=3i^+5j^+2k^$$      , Q→$$=5i^$$−$$j^+4k^$$      and R→$$=4i^+2j^+3k^$$     are coplanar, then:

Infinity Learn · Accepted Answer

The  scalar triple product P→.(Q$$→×R→$$)=(P$$→×Q→$$).R→      for three coplanar  vectors is always zero, as it represents the volume of parallelepiped formed  with three adjacent sides as the vectors P→,Q→      and R→     .  Here  P→×Q→$$=(3i^+5j^+2k^)$$×(5i^$$−$$j^+4k^)     $$=$$| $$i^$$     $$j^$$     $$k^$$  $$3$$     $$5$$     $$2$$  $$5$$   −$$1$$    $$4$$ |     $$=i^(20+2)$$−$$j^(12$$−$$10)+k^($$−$$3$$−$$25)$$     $$=22i^$$−$$2j^$$−$$28k^$$     (P$$→×Q→$$).R→$$=(22i^$$−$$2j^$$−$$28k^)$$.(4i+2j^+3k^)     $$=88$$−$$4$$−$$84=zero$$     .

If $\vec{P} = 3 \hat{i} + 5 \hat{j} + 2 \hat{k}$ , $\vec{Q} = 5 \hat{i} - \hat{j} + 4 \hat{k}$ and $\vec{R} = 4 \hat{i} + 2 \hat{j} + 3 \hat{k}$ are coplanar, then:

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If P→=3i^+5j^+2k^ , Q→=5i^−j^+4k^ and R→=4i^+2j^+3k^ are coplanar, then:

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If $\vec{P} = 3 \hat{i} + 5 \hat{j} + 2 \hat{k}$ , $\vec{Q} = 5 \hat{i} - \hat{j} + 4 \hat{k}$ and $\vec{R} = 4 \hat{i} + 2 \hat{j} + 3 \hat{k}$ are coplanar, then: