A golfer, putting on a green requires three strokes to “hole the ball.” During the first putt, the ball roles 5.0m due east. For the second putt, the ball travels 2.1m at an angle of 20 degrees north of east. The third putt is 0.50m due north. What displacement (magnitude and direction relative to due east) would have been needed to “hole the ball” on the very first putt? Use components to solve this problem.

Identify the three vectors?
Sketch the vectors and show the vector sum?
Identify the components of the three vectors?
Ax= Ay=
Bx= By=
Cx= Cy=
Determine the components of the resultant vector
Sx= Sy=
Convert this into the magnitude and direction of the resultant vector
IsI =
Angle =

To solve this problem, let's break it down into the different components and vectors involved.

First, let's identify the three vectors:
1) The first putt that rolls 5.0m due east.
2) The second putt that travels 2.1m at an angle of 20 degrees north of east.
3) The third putt that is 0.50m due north.

Now, let's sketch the vectors on a flat surface like a piece of paper, making sure to indicate the direction and magnitude of each vector.

For the components of each vector:
1) The first putt: Since it is due east, there is no north component (Ay = 0), and the east component is equal to the magnitude of the vector (Ax = 5.0m).
2) The second putt: To find the components, we need to calculate the east and north components using the angle and magnitude. East component (Bx) can be found by Bx = magnitude * cos(angle) = 2.1m * cos(20 degrees). The north component (By) can be found by By = magnitude * sin(angle) = 2.1m * sin(20 degrees).
3) The third putt: Since it is due north, there is no east component (Cx = 0), and the north component is equal to the magnitude of the vector (Cy = 0.50m).

Let's calculate the components:
Ax = 5.0m (from the first putt)
Ay = 0m (no north component for the first putt)
Bx = 2.1m * cos(20 degrees) ≈ 1.970m (east component from the second putt)
By = 2.1m * sin(20 degrees) ≈ 0.713m (north component from the second putt)
Cx = 0m (no east component for the third putt)
Cy = 0.50m (north component from the third putt)

Now, let's add up the components to find the resultant vector:
Sx = Ax + Bx + Cx = 5.0m + 1.970m + 0m = 6.970m
Sy = Ay + By + Cy = 0m + 0.713m + 0.50m ≈ 1.213m

The components of the resultant vector are Sx = 6.970m and Sy = 1.213m.

To convert these components into magnitude and direction:
Magnitude of the resultant vector, |S|, can be found using the Pythagorean theorem: |S| = sqrt(Sx^2 + Sy^2).
Angle, θ, can be found using the inverse tangent function: θ = atan(Sy/Sx).

Calculating the magnitude and angle:
|S| = sqrt((6.970m)^2 + (1.213m)^2) ≈ 7.082m
θ = atan(1.213m / 6.970m) ≈ 9.886 degrees

Therefore, the magnitude of the resultant vector is approximately 7.082m, and the direction relative to due east is approximately 9.886 degrees.