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If u, v, and w are three vectors that are not in the same plane, then the value of (u+v-w)· ((u-v)×(v-w)) is

1. 3 u·(v× w)
2. 0
3. u·(v× w)
4. u·(w× v)

Correct answer: u·(v× w)

Solution

The expression (u+v-w) represents a linear combination of the vectors, and when dotted with the cross product ((u-v) imes(v-w)), it simplifies to the scalar triple product, which is equal to uullet(v imes w). This is because the scalar triple product gives the volume of the parallelepiped formed by the three vectors, and since they are not coplanar, this value is non-zero.

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