Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 9, Problem 67PQ

(a)

To determine

The angle between the pair of vectors.

(a)

Expert Solution
Check Mark

Answer to Problem 67PQ

The angle between the pair of vectors is 90°_.

Explanation of Solution

Write the expression for the magnitude of vector A.

    A=Ax2+Ay2+Az2 (I)

Here, Ax is the x component of A, Ay is the y component of A and Az is the z component of A.

Write the expression for the magnitude of vector B.

    B=Bx2+By2+Bz2 (II)

Here, Bx is the x component of B, By is the y component of B and Bz is the z component of B.

Write the expression for dot product between A and B.

    AB=AxBx+AyBy+AzBz (III)

Write the expression for angle between A and B.

    θ=cos1(ABAB) (IV)

Conclusion:

Substitute 3.00 for Ax, 1.00 for Ay and +4.00 for Az in equation (I).

    A=(3.00)2+(1.00)2+(+4.00)2=5.10

Substitute +2.00 for Bx, +2.00 for By and +2.00 for Bz in equation (II).

    B=(+2.00)2+(+2.00)2+(+2.00)2=3.46

Substitute 3.00 for Ax, 1.00 for Ay, +4.00 for Az, +2.00 for Bx, +2.00 for By and +2.00 for Bz in equation (III).

    AB=(3.00)(+2.00)+(1.00)(+2.00)+(+4.00)(+2.00)=0

Substitute 0 for AB, 5.10 for A and 3.46 for B in equation (IV).

    θ=cos1(0(5.10)(3.46))=90°

Therefore, the angle between the pair of vectors is 90°_.

(b)

To determine

The angle between the pair of vectors.

(b)

Expert Solution
Check Mark

Answer to Problem 67PQ

The angle between the pair of vectors is 120°_.

Explanation of Solution

Write the expression for the magnitude of vector A.

    A=Ax2+Ay2+Az2 (I)

Here, Ax is the x component of A, Ay is the y component of A and Az is the z component of A.

Write the expression for the magnitude of vector B.

    B=Bx2+By2+Bz2 (II)

Here, Bx is the x component of B, By is the y component of B and Bz is the z component of B.

Write the expression for dot product between A and B.

    AB=AxBx+AyBy+AzBz (III)

Write the expression for angle between A and B.

    θ=cos1(ABAB) (IV)

Conclusion:

Substitute +1.00 for Ax, +2.00 for Ay and 0 for Az in equation (I).

    A=(+1.00)2+(+2.00)2+(0)2=2.24

Substitute 0 for Bx, 2.00 for By and 3.00 for Bz in equation (II).

    B=(0)2+(2.00)2+(3.00)2=3.61

Substitute  +1.00 for Ax, +2.00 for Ay, 0 for Az, 0 for Bx, 2.00 for By and 3.00 for Bz in equation (III).

    AB=(+1.00)(0)+(+2.00)(2.00)+(0)(3.00)=4.00

Substitute 4.00 for AB, 2.24 for A and 3.61 for B in equation (IV).

    θ=cos1(4.00(2.24)(3.61))=120°

Therefore, the angle between the pair of vectors is 120°_.

(c)

To determine

The angle between the pair of vectors.

(c)

Expert Solution
Check Mark

Answer to Problem 67PQ

The angle between the pair of vectors is 86°_.

Explanation of Solution

Write the expression for the magnitude of vector A.

    A=Ax2+Ay2+Az2 (I)

Here, Ax is the x component of A, Ay is the y component of A and Az is the z component of A.

Write the expression for the magnitude of vector B.

    B=Bx2+By2+Bz2 (II)

Here, Bx is the x component of B, By is the y component of B and Bz is the z component of B.

Write the expression for dot product between A and B.

    AB=AxBx+AyBy+AzBz (III)

Write the expression for angle between A and B.

    θ=cos1(ABAB) (IV)

Conclusion:

Substitute +4.00 for Ax, 0 for Ay and +2.00 for Az in equation (I).

    A=(+4.00)2+(0)2+(+2.00)2=4.47

Substitute 1.00 for Bx, +5.00 for By and +3.00 for Bz in equation (II).

    B=(1.00)2+(+5.00)2+(+3.00)2=5.92

Substitute +4.00 for Ax, 0 for Ay, +2.00 for Az, 1.00 for Bx, +5.00 for By and +3.00 for Bz in equation (III).

    AB=(+4.00)(1.00)+(0)(+5.00)+(+2.00)(+3.00)=2.00

Substitute 0 for AB, 4.47 for A and 5.92 for B in equation (IV).

    θ=cos1(2.00(4.47)(5.92))=86°

Therefore, the angle between the pair of vectors is 86°_.

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Chapter 9 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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