Differential Equations and Linear Algebra (4th Edition)
Differential Equations and Linear Algebra (4th Edition)
4th Edition
ISBN: 9780321964670
Author: Stephen W. Goode, Scott A. Annin
Publisher: PEARSON
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Chapter 5.5, Problem 1AP

For Problem 1-2, determine the angle between the given vectors u and v using the standard inner product on n .

u = ( 2 , 3 ) and v = ( 4 , 1 )

Expert Solution & Answer
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To determine

The angle between the given vectors u and v.

Answer to Problem 1AP

Solution:

The angle between the vectors is θ=cos1(5221)1.23rad.

Explanation of Solution

Given:

The vectors are u=(2,3) and v=(4,1)

Approach:

The angle between two non zero vectors u and v is given by,

cosθ=uvuv(1)

If u=(a1,a2.a3) and v=(b1,b2,b3) are vectors, then the dot product of u and v is denoted by uv, and defined by,

uv=a1b1+a2b2+a3b3(2)

The magnitude of the vector u=(a1,a2.a3) is given by,

u=a12+a22+a32(3)

Calculation:

Substitute (2,3) for u and (4,1) for v in equation (2)

uv=2(4)+3(1)=83=5

Substitute (2,3) for u in equation (3)

u=22+32=4+9=13

Substitute (2,3) for v in equation (3).

v=42+(1)2=16+1=17

Substitute 5 for uv, 13 for u and 17 for v in equation (1).

cosθ=51317θ=cos15221

Therefore, the angle between the vectors is θ=cos1(5221)1.23rad.

Conclusion:

Hence, the angle between the vectors is θ=cos1(5221)1.23rad.

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Chapter 5 Solutions

Differential Equations and Linear Algebra (4th Edition)

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