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Let P=(2,3,0,-1)and Q=(3,-2,1,6)be points in four-dimensional space.

(a) Find PQโ†’.

(b) Find PQOโ†’.

(c) Find the unit vector in the direction of PQโ†’.

Short Answer

Expert verified

Part A:

The value of PQโ†’is โŸจ1,-5,1,7โŸฉ

Part B:

The norm of the vector โ€–PQโ†’โ€–is 76.

Part C:

The unit vector in the direction of PQโ†’is

176โŸจ1,-5,1,7โŸฉ

Step by step solution

01

Introduction (Part A)

  • Consider the vector P=(2,3,0,-1)and Q=(3,-2,1,6)be the points in four-dimensional space.
  • The Target is to find PQโ†’.
02

Given Information (Part A)

If there are two points to consider

P=x0,y0,z0,w0and Q=x1,y1,z1,w1,

then

PQโ†’=x1-x0,y1-y0,z1-z0,w1-w0.

03

Explanation (Part A)

Now,

P=(2,3,0,-1)and Q=(3,-2,1,6)

As a result , PQโ†’=โŸจ3-2,-2-3,1-0,6-(-1)โŸฉ

Hence, PQโ†’=โŸจ1,-5,1,7โŸฉ

As a result, the value of PQโ†’is โŸจ1,-5,1,7โŸฉ

04

Introduction (Part B)

  • Consider the points P=(2,3,0,-1)and Q=(3,-2,1,6)be the points in four-dimensional space.
  • The aim is to find the norm โ€–PQโ†’โ€–.
05

Given Information (Part B)

If

vis the vector in R4such that v=โŸจa,b,c,dโŸฉ,

then

norm of the vector is given by

โ€–vโ€–=a2+b2+c2+d2

06

Explanation (Part B)

Now, P=(2,3,0,-1)andQ=(3,-2,1,6)

Therefore,PQโ†’=โŸจ3-2,-2-3,1-0,6-(-1)โŸฉ

Hence,PQโ†’=โŸจ1,-5,1,7โŸฉ

Therefore,

โ€–PQโ†’โ€–=12+(-5)2+12+72

=1+25+1+49

=76

Therefore, the norm of the vectorโ€–PQโ†’โ€–is76

07

Introduction (Part C)

  • Consider the points P=(2,3,0,-1)and Q=(3,-2,1,6)be the points in four - dimensional space.
  • The objective is to find the unit vector in the direction ofPQโ†’
08

Given Information(Part C)

Now, P=(2,3,0,-1)and Q=(3,-2,1,6)

Therefore,PQโ†’=โŸจ3-2,-2-3,1-0,6-(-1)โŸฉ

Hence, PQโ†’=โŸจ1,-5,1,7โŸฉ.

09

Explanation (Part C)

โ€–PQโ†’โ€–=12+(-5)2+12+72

=1+25+1+49

=76

10

Explanation (Part C)

The unit vector in the direction of PQยฏis given by 1โ€–PQโ†’โ€–PQโ†’.

Substitute the values,

PQโ†’=176โŸจ1,-5,1,7โŸฉ

As a result, the unit vector in the direction of

PQโ†’is 176โŸจ1,-5,1,7โŸฉ

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