<p>If \(\vec{u}\), \(\vec{v}\) and \(\vec{w}\) are three non-coplanar vectors, then \((\vec{u}+\vec{v}-\vec{w})\cdot(\vec{u}-\vec{v})\times(\vec{v}-\vec{w})\) equals</p>
<p>\(0\)</p>
<p>\(\vec{u}\cdot\vec{v}\times\vec{w}\)</p>
<p>\(\vec{u}\cdot\vec{w}\times\vec{v}\)</p>
<p>\(3\vec{u}\cdot\vec{v}\times\vec{w}\)</p>
Step-by-Step Solution
Key Concept: Use the scalar triple product property that [a, b, c] = a·(b×c) is invariant under cyclic permutations and linear combinations of vectors in specific ways. Expand the cross product first, then apply the distributive property of dot product with scalar triple products.
Step 1: Expand the cross product (u-v)×(v-w): (u-v)×(v-w) = u×v - u×w - v×v + v×w = u×v - u×w + v×w Step 2: Compute the dot product (u+v-w)·(u×v - u×w + v×w): = u·(u×v) - u·(u×w) + u·(v×w) + v·(u×v) - v·(u×w) + v·(v×w) - w·(u×v) + w·(u×w) - w·(v×w) Step 3: Apply the property that a·(a×b) = 0 for any vectors a, b: Terms u·(u×v) = 0, v·(u×v) = 0, v·(v×w) = 0, w·(u×w) = 0, w·(v×w) = 0 Step 4: Remaining terms are: u·(v×w) - v·(u×w) - w·(u×v) = [u,v,w] - [v,u,w] - [w,u,v] = [u,v,w] + [u,v,w] - [u,v,w] = [u,v,w] Since u, v, w are non-coplanar, [u,v,w] ≠ 0 ∴ Answer: B
Correct Answer: B