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Worked solution

Factor a cross product to get a linear relation

Three non-zero vectors a\mathbf a, b\mathbf b and c\mathbf c are such that 2a×b=a×3c2\mathbf a\times\mathbf b=\mathbf a\times3\mathbf c.

Find a linear relationship between a\mathbf a, b\mathbf b and c\mathbf c.

CJC · Discussion · Question 8

Answer

2b−3c=ka2\mathbf b-3\mathbf c=k\mathbf a for some k∈Rk\in\mathbb R.

Factor the common cross-product argument

2a×b−a×3c=0,a×(2b−3c)=0.\begin{aligned}2\mathbf a\times\mathbf b-\mathbf a\times 3\mathbf c&=\mathbf 0,\\\mathbf a\times(2\mathbf b-3\mathbf c)&=\mathbf 0.\end{aligned}

Since a≠0\mathbf a\ne\mathbf 0, the vector 2b−3c2\mathbf b-3\mathbf c is a scalar multiple of a\mathbf a. Thus

2b−3c=ka,k∈R.2\mathbf b-3\mathbf c=k\mathbf a,\qquad k\in\mathbb R.

This includes the case 2b−3c=02\mathbf b-3\mathbf c=\mathbf 0 with k=0k=0, even though b and c individually are nonzero.

How did this part go?

Common mistakes

Do not “cancel a” from a cross product as though it were ordinary multiplication. The scalar k may be zero.

Solution notes

The real-number symbol omitted by extraction is visibly ℝ in the source. No source mathematical correction is needed.

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