The Scalar ProductThe vectors A andB are given by A = 31 + 1j and B = -1î + 1j.(a) Determine the scalar product A - B.SOLUTIONConceptualize There is no physical system to imagine here. Rather, it is purely a mathematical exercise involving two (vectorsCategorize Because we have a definition for the scalar product, we categorize this example as a substitution vy problem.Substitute the specific vector expressions for A and B:A-B = (31 + 1j) - (-1î + 1j)= [3i - (-1)i - [31 - 15 - 13 - (-1) + [1j - 15]The same result is obtained when we use the equation A -B = AB, + A8, + A,B, directly, where A, = 3, A, = 1, B, = -1 and 8, = 1.(b) Find the angle e between A and B.SOLUTIONEvaluate the magnitudes of A and B using the Pythagorean theorem:A = VA,? + A, =B = V82 + B 2 =Use the equation A - B = AB cos(e) and the result from part (a) to find the angle (in degrees).cos(e) =ABEXERCISEAs 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.)HintActivate Windows

(a) The scalar product is, A→·B→=3i+1j·-1i+1jA→·B→=3i·-1i+3i·1 j+1j·-1i+1j·1jA→·B→=-3+0+0+1=-2

The Scalar Product The vectors A and are given by A= 3i + 1j and B = -1i - 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 Categorize Because we have a definition for the scalar product, we categorize this example as a substitution vY problem. Substitute the specific vector expressions for A and B: A-B- (3i - 1j) - (-1i + 1j) = [31-(-1)1 - [3i - 1- [3 - (-1 - [13 - 15 The same result is obtained when we use the equation A - B = A8, - A8, + A8, directly, where A, = 3, A, = 1. 8, = -1 and 8, = 1. (b) Find the angle e between A and B. SOLUTION Evaluate the magnitudes of A and B using the Pythagorean theorem: A- VAA2 = B = V8,2 +82 = Use the equation A-B = AB cos(e) and the result from part (a) to find the angle (in degrees). A-B cos(e)= AB - cos EXERCISE As a particle moves from the origin to (7i - 5j) 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 1.)

The Scalar Product The vectors A and are given by A= 3i + 1j and B = -1i - 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 Categorize Because we have a definition for the scalar product, we categorize this example as a substitution vY problem. Substitute the specific vector expressions for A and B: A-B- (3i - 1j) - (-1i + 1j) = [31-(-1)1 - [3i - 1- [3 - (-1 - [13 - 15 The same result is obtained when we use the equation A - B = A8, - A8, + A8, directly, where A, = 3, A, = 1. 8, = -1 and 8, = 1. (b) Find the angle e between A and B. SOLUTION Evaluate the magnitudes of A and B using the Pythagorean theorem: A- VAA2 = B = V8,2 +82 = Use the equation A-B = AB cos(e) and the result from part (a) to find the angle (in degrees). A-B cos(e)= AB - cos EXERCISE As a particle moves from the origin to (7i - 5j) 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 1.)