Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

For any vectors \({\rm{u,v}}\)and \(w\)in \({{\rm{V}}_{\rm{3}}}\),

\({\rm{u}} \cdot \left( {{\rm{v \times w}}} \right){\rm{ = }}\left( {{\rm{u \times v}}} \right) \cdot {\rm{w}}\)

Short answer

The stated statement is true.

Step-by-step solution

Look at the left-hand side.

If \({\rm{u = < a,b,c > ,v = < d,e,f > ,w = < q,r,s > }}\)then

\(\begin{aligned}{c}{\rm{u}} \cdot {\rm{(v \times w) = < a,b,c > }} \cdot {\rm{ < es - fr,fq - ds,dr - eq > }}\\{\rm{ = a(es - fr) + b(fq - ds) + c(dr - eq)}}\end{aligned}\)

Look at the right hand side.

\(\begin{aligned}{c}{\rm{(u \times v)}} \cdot {\rm{w = < bf - ce,cd - af,ae - bd > }} \cdot {\rm{ < q,r,s > }}\\{\rm{ = q(bf - ce) + r(cd - af) + s(ae - bd)}}\\{\rm{ = qbf - qce + rcd - raf + sae - sbd}}\\{\rm{ = (sae - raf) + (qbf - sbd) + (rcd - qce)}}\\{\rm{ = a(es - fr) + b(fq - ds) + c(dr - eq)}}\end{aligned}\)

L.H.S=R.H.S

Therefore,, the stated statement is true.