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If u and v are nonzero vectors in 3, why do the equations role="math" localid="1649263352081" u·(u×v)=0 and v·(u×v)=0 tell us that the cross product is orthogonal to both u and v?

Short Answer

Expert verified

We prove thatu·(u×v)=0andv·(u×v)=0tell us that the cross product is orthogonal to both u and v.

Step by step solution

01

Step 1. Given Information 

If u and v are nonzero vectors in 3, why do the equations u·(u×v)=0 and v·(u×v)=0 tell us that the cross product is orthogonal to both u andv?

02

Step 2. u and v are nonzero vectors in ℝ3.

Let u=(u1,u2,u3)andv=(v1,v2,v3).

The determinant of a 3 × 3 matrix is

u×v=detijku1u2u3v1v2v3

03

Step 3. Then, by the definition of the cross product,

u×v=(u2v3u3v2,u3v1u1v3,u1v2u2v1)

Now proving the equation u·(u×v)=0

u·(u×v)=(u1,u2,u3)·(u2v3u3v2,u3v1u1v3,u1v2u2v1)u·(u×v)=u1(u2v3u3v2)+u2(u3v1u1v3)+u3(u1v2u2v1)u·(u×v)=0

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Step 4. Now proving the equationv·(u×v)=0.

v·(u×v)=(v1,v2,v3)·(u2v3u3v2,u3v1u1v3,u1v2u2v1)v·(u×v)=v1(u2v3u3v2)+v2(u3v1u1v3)+v3(u1v2u2v1)v·(u×v)=0

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