Aforce of 8 n makes an angle of π/4 radian with the y-axis, pointing to the right. the force acts against the movement of an object along the straight line connecting (1, 3) to (4, 5).

a. find a formula for the force vector f

b. find the angle θ between the displacement direction d = (4 − 1)i + (5 − 3)j and the force direction f. (round your answer to one decimal place.)

c. the work done is f · d or, equivalently, ||fd||cos(θ). compute the work from both formulas and compare

a) F= 4i + 4j

b) Θ=11.3

c) The work done is 20

Step-by-step explanation:

a) ||F||=8, α=π/4

Fx=||F||·sin(π/4)=8·

Fy=||F||·cos(π/4)=8·

F=Fx i + Fy j = 4i + 4j

b) We can find the value of Θ using the equation:

cos(Θ)=

where:

D= 3 i + 2 j

F=4i + 4j

The dot product is defined as the sum of the products of the components of each vector as:

F · D=

||F||= 8

||D||=

Hence:

Θ=arccos()

Θ=arccos(0.981)

Θ= 11.3°

c) Work is equal to:

F · D= =28.3

Other way of obtainig the work is:

||F||||D||cos(Θ)

where:

||F||= 8

||D||=

Θ=11.3°

So,  ||F||||D||cos(Θ)=8××cos(11.3°)=28.3

the answer is 62.8 cubic units