(b) Find the angle e between A and B. SOLUTION Evaluate the magnitudes of A and B using the Pythagorean theorem: A = VAA - A Use the equation A.B AB cos(e) and the result from part (a) to find the angle (in degrees). cos(6) = cos - 116.6 EXERCISE As a particle moves from the origin to (7i - 55) m, it is acted upon by a force given by (7i + 75) N. Calculate the work done by this force on the particle as it moves through the given displacement. (Give your answer in J.) Hint W = 14 Use the expression for the work done by a force acting on an object as it is displaced to calculate the work done. 3
(b) Find the angle e between A and B. SOLUTION Evaluate the magnitudes of A and B using the Pythagorean theorem: A = VAA - A Use the equation A.B AB cos(e) and the result from part (a) to find the angle (in degrees). cos(6) = cos - 116.6 EXERCISE As a particle moves from the origin to (7i - 55) m, it is acted upon by a force given by (7i + 75) N. Calculate the work done by this force on the particle as it moves through the given displacement. (Give your answer in J.) Hint W = 14 Use the expression for the work done by a force acting on an object as it is displaced to calculate the work done. 3
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Transcribed Image Text:The Scalar Product
The vectors A and B are given by A = 31 + lj and B = -1î + 1j.
(a) Determine the scalar product A- B.
SOLUTION
Conceptualize There is no physical system to imagine here. Rather, it is purely a mathematical exercise involving two (vectors v
Categorize Because we have a definition for the scalar product, we categorize this example as (a substitution v problem.
Substitute the specific vector expressions for A and B:
A-B - (31 + 1j) - (-1i + 1j)
- [3i - (-1)1 - [31 - 15 - [13 - (-1)1 + [19 - 15
= |-2
The same result is obtained when we use the equation A- B = A B + A B +AB, directly, where A, = 3, A, = 1, B, = -1 and B = 1.
(b) Find the angle e between A and B.
SOLUTION
Evaluate the magnitudes of A and B using the Pythagorean theorem:
A = VA? + A 2 = A
B = VB 2 +B 2 =0
Use the equation A - B = AB cos(e) and the result from part (a) to find the angle (in degrees).
A-B
cos(e) =
= 116.8
AB
EXERCISE
As a particle moves from the origin to (7i + 5j) m, it is acted upon by a force given by (7î + 7j) N. Calculate the work done by this force on the particle as it moves through the given displacement. (Give your answer in J.)
Hint
W = 14
Use the expression for the work done by a force acting on an object as it is displaced to calculate the work done. J
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