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Let u,v, and wbe three mutually perpendicular vectors in 3.

(a) Prove that u×(v×w)=0.

(b) Show that |u·(v×w)|=uvw.

Short Answer

Expert verified

Ans:

part (a).

u×(v×w)=uwcos90°v-uvcos90°w=(0)v-(0)w=0

part (b)

.u·(v×w)=uv×wcosθ=uv×wcos0°=uv×w=uvwsin90°=uvw

Step by step solution

01

Step 1. Given information:

There are three mutually perpendicular vectors in 3.

  • u×(v×w)=0
  • |u·(v×w)|=uvw
02

Step 2. Solving part (a):

The equation u×(v×w)=(u·w)v-(u·v)wgives:

u×(v×w)=uwcos90°v-uvcos90°w(Vectors are perpendicular)=(0)v-(0)w(Simplify)=0

Therefore, if three vectors are mutually perpendicular then u×(v×w)=0holds.

03

Step 3. Solving part (b).

The vectors uand v×ware parallel to each other.

The value of u·(v×w)is:

role="math" localid="1650748918202" u·(v×w)=uv×wcosθ=uv×wcos0°(Angle betweenuandv×w=uv×w=uvwsin90°(Because vectorsvandware perpendicular)=uvw

Therefore, the result u·(v×w)=uvw is proved.

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