Porisma

Worked solution

Use a zero cross product to prove parallelism

Given a×b=c×a\mathbf a\times\mathbf b=\mathbf c\times\mathbf a, ∣a∣≠0|\mathbf a|\ne0, and nonparallel vectors b,c\mathbf b,\mathbf c, show that a\mathbf a is parallel to b+c\mathbf b+\mathbf c.

CJC · Vectors Basic · Example 16

Answer

a∥(b+c)\mathbf a\parallel(\mathbf b+\mathbf c).

Compare your reasoning

Move both cross products to one side

a×b=c×a⟹(a×b)−(c×a)=0\mathbf a\times\mathbf b=\mathbf c\times\mathbf a\quad\Longrightarrow\quad(\mathbf a\times\mathbf b)-(\mathbf c\times\mathbf a)=\mathbf0

Keep their order until applying the sign change.

Reverse one product then distribute

(a×b)−[−(a×c)]=0,(a×b)+(a×c)=0,a×(b+c)=0.\begin{aligned}(\mathbf a\times\mathbf b)-[-(\mathbf a\times\mathbf c)]&=\mathbf0,\\(\mathbf a\times\mathbf b)+(\mathbf a\times\mathbf c)&=\mathbf0,\\\mathbf a\times(\mathbf b+\mathbf c)&=\mathbf0.\end{aligned}

Anti-commutativity turns subtraction into addition.

Check both factors are nonzero

a≠0,b+c≠0⟹a∥(b+c)\mathbf a\ne\mathbf0,\quad\mathbf b+\mathbf c\ne\mathbf0\quad\Longrightarrow\quad\mathbf a\parallel(\mathbf b+\mathbf c)

a is nonzero by hypothesis. If b+c=0, then c=−b, contradicting the given nonparallel directions. A zero cross product of these two nonzero vectors therefore proves parallelism.

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