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Let u, v, andw be vectors in 3. Prove that u×v=u×wif and only if u is parallel to vw.

Short Answer

Expert verified

Hence, we prove thatu×v=u×w if and only if u is parallel to v-w.

Step by step solution

01

Step 1. Given Information

Let u, v, and w be vectors in 3. Prove that u×v=u×w if and only if u is parallel to vw.

02

Step 2. Let u, v, and w be vectors in ℝ3.

u×v=u×wSubtractu×wonbothsideu×v-u×w=u×w-u×wu×v-u×w=0u×(v-w)=0u=0v-w=0

That meansuv-w.

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