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Let u and v be vectors in 3 such that u·v=0. Prove that if θ is the angle between u and v, then role="math" localid="1650011942678" tanθ=u×vu·v

Short Answer

Expert verified

Hence, we prove that if θ is the angle between u and v, then tanθ=u×vu·v

Step by step solution

01

Step 1. Given Information

Let u and v be vectors in 3 such that u·v=0. Prove that if θ is the angle between u and v, then tanθ=u×vu·v

02

Step 2. As we know that if u, v, and u × v form a right-handed triple.

Then, u×v=uvsinθEquation1

u, v, and u·v form a left-handed triple.

role="math" localid="1650014121047" u·v=uvcosθEquation2

03

Step 3. Divide the equation 1 and 2

u×vu·v=uvsinθuvcosθu×vu·v=sinθcosθu×vu·v=tanθtanθ=u×vu·v

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