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Chapter 10: Q. 72 (page 826)

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

(a) Prove that u脳(v脳w)=0.

(b) Show that |u路(v脳w)|=鈥u鈥鈥v鈥鈥w鈥.

Short Answer

Expert verified

Ans:

part (a).

u脳(v脳w)=鈥u鈥鈥w鈥cos90掳v-鈥u鈥鈥v鈥cos90掳w=(0)v-(0)w=0

part (b)

.u路(v脳w)=鈥u鈥鈥v脳w鈥cos胃=鈥u鈥鈥v脳w鈥cos0掳=鈥u鈥鈥v脳w鈥=鈥u鈥鈥v鈥鈥w鈥sin90掳=鈥u鈥鈥v鈥鈥w鈥

Step by step solution

01

Step 1. Given information:

There are three mutually perpendicular vectors in 鈩3.

  • u脳(v脳w)=0
  • |u路(v脳w)|=鈥u鈥鈥v鈥鈥w鈥
02

Step 2. Solving part (a):

The equation u脳(v脳w)=(u路w)v-(u路v)wgives:

u脳(v脳w)=鈥u鈥鈥w鈥cos90掳v-鈥u鈥鈥v鈥cos90掳w(Vectors are perpendicular)=(0)v-(0)w(Simplify)=0

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

03

Step 3. Solving part (b).

The vectors uand v脳ware parallel to each other.

The value of u路(v脳w)is:

role="math" localid="1650748918202" u路(v脳w)=鈥u鈥鈥v脳w鈥cos胃=鈥u鈥鈥v脳w鈥cos0掳(Angle betweenuandv脳w=鈥u鈥鈥v脳w鈥=鈥u鈥鈥v鈥鈥w鈥sin90掳(Because vectorsvandware perpendicular)=鈥u鈥鈥v鈥鈥w鈥

Therefore, the result u路(v脳w)=鈥u鈥鈥v鈥鈥w鈥 is proved.

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Most popular questions from this chapter

In Exercises 30鈥35 compute the indicated quantities when u=(鈭3,1,鈭4),v=(2,0,5),andw=(1,3,13)

role="math" localid="1649436488889" (u脳v)路wandu路(v脳w)

Why do we use the terminology "separable" to describe a differential equation that can be written in the form

Find PQ鈬赌

P=5,-3,Q=3,-6

Fill in the blanks to complete each of the following theorem statements:

For 蔚>0,f(x)鈭(L-蔚,L+蔚)if and only if<蔚.

Find PQ鈫

P=-1,5,4,Q=-1,6,4
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