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Question

Let u and v are unit vectors and w is a vector such that u×v+u=w and w×u=v then find the value of [uvw].


A

0

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B

1

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C

|w|

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D

2|w|

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Solution

The correct option is B

1


Explanation for the correct option:

Given that, u×v+u=w

Multiplying both the sides with vector u

(u×v+u)×u=w×u(u×v)×u+u×u=v(u·u)v(v·u)u+u×u=v(1)v(v·u)u+0=vv(v·u)u=vu·v=0

Now,

u·(v×w)=u·(v×(u×v+u))=u·(v×(u×v)+v×u)=u·((v·v)u(v·u)v+v×u)=u(|v|2u-0+v×u)=|v|2(u·u)u·(v×u)=|v|2|u|20=1

Therefore, [uvw]=1

Hence, the correct option is (B)


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