PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter 6, Problem 14RE
Solution

a)

To find: the direction angles of u and v.

The direction angle of u=2,4 is 116.51° or 2.03 radians and the direction angle of v=6,4 is 33.69° or 0.59 radians.

Given information:

Given two vectors u=2,4,

  v=6,4 .

Formula used:

  • If θ is the direction angle of v=x,y, then x=vcosθ, y=vcosθ .
  • The length of the vector v=x,y is v=x2+y2 .

Calculation:

The length of the vector u=2,4:

Substituting 2 for x and 4 for y in x2+y2, the length of the vector u is 22+42=4+16=20=25

Thus, the length of the vector u is |u|=25 .

The length of the vector v=6,4:

Substituting 6 for x and 4 for y in x2+y2, the length of the vector v is 62+42=36+16=52=213

Thus, the length of the vector v is |v|=213 .

The direction angle of u=2,4:

Substituting 2 for x and 25 for u in x=ucosθ, then

  cosθ=225θ=cos115116.51°

The direction angle of u=4,3 is 116.51° or 2.03 radians.

The direction angle of v=6,4:

If θ is the direction angle of v, then 6=vcosθ . Thus, θ=cos16v .

By substituting 213 for v , the required angle computed as follows:

  θ=cos1621333.69°

The direction angle of v=6,4 is 33.69° or 0.59 radians.

  PRECALCULUS:GRAPHICAL,...-NASTA ED., Chapter 6, Problem 14RE

b)

To find: the angle between the vectors u and v.

The angle between the vectors u=2,4 and v=6,4 is 82.88° or 1.45 radians.

Given information:

Given two vectors u=2,4,v=6,4.

From part (a), the length of the vectors u and v respectively are 25 and 213.

Formula used:

  • If θ is the angle between the nonzero vectors u and v , then cosθ=u.vuv .
  • The dot product of two vectors u=x1,y1,v=x2,y2 is defined as u.v=x1x2+y1y2.

Calculation:

The dot product of uand v :

The dot product of two vectors u=2,4,v=6,4 is computed as follows:

  u.v=26+44=12+16=4.

Angle between the vectors:

If θ is the angle between the vectors u and v , then by substituting u.v=4,|u|=25 and |v|=213 , the value of cosθ is given below:

  cosθ=u.vuv=425213=165.

Thus, the required angle is

  θ=cos116582.88°

Therefore, the angle between the vectors u=2,4 and v=6,4 is 82.88° or 1.45 radians.

Chapter 6 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. 6.1 - Prob. 1ECh. 6.1 - Prob. 2ECh. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.1 - Prob. 8ECh. 6.1 - Prob. 9ECh. 6.1 - Prob. 10ECh. 6.1 - Prob. 11ECh. 6.1 - Prob. 12ECh. 6.1 - Prob. 13ECh. 6.1 - Prob. 14ECh. 6.1 - Prob. 15ECh. 6.1 - Prob. 16ECh. 6.1 - Prob. 17ECh. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Prob. 21ECh. 6.1 - Prob. 22ECh. 6.1 - Prob. 23ECh. 6.1 - Prob. 24ECh. 6.1 - Prob. 25ECh. 6.1 - Prob. 26ECh. 6.1 - Prob. 27ECh. 6.1 - Prob. 28ECh. 6.1 - Prob. 29ECh. 6.1 - Prob. 30ECh. 6.1 - Prob. 31ECh. 6.1 - Prob. 32ECh. 6.1 - Prob. 33ECh. 6.1 - Prob. 34ECh. 6.1 - Prob. 35ECh. 6.1 - Prob. 36ECh. 6.1 - Prob. 37ECh. 6.1 - Prob. 38ECh. 6.1 - Prob. 39ECh. 6.1 - Prob. 40ECh. 6.1 - Prob. 41ECh. 6.1 - Prob. 42ECh. 6.1 - Prob. 43ECh. 6.1 - Prob. 44ECh. 6.1 - Prob. 45ECh. 6.1 - Prob. 46ECh. 6.1 - Prob. 47ECh. 6.1 - Prob. 48ECh. 6.1 - Prob. 49ECh. 6.1 - Prob. 50ECh. 6.1 - Prob. 51ECh. 6.1 - Prob. 52ECh. 6.1 - Prob. 53ECh. 6.1 - 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