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 =

On a safari, a team of naturalists sets out toward a research station located 7.0 km away in a direction 42° north of east. After traveling in a straight line for 3.1 km, they stop and discover that they have been traveling 20° north of east, because their guide misread his compass. What is direction (relative to due east) of the displacement vector now required to bring the team to the research station?

To find the displacement needed for the first putt, we need to analyze the given vectors and their components. Let's go through the steps:

1. Identify the three vectors:
- The first putt travels 5.0m due east.
- The second putt travels 2.1m at an angle of 20 degrees north of east.
- The third putt travels 0.50m due north.

2. Sketch the vectors and show the vector sum:
It can be helpful to sketch the vectors on a coordinate plane. Draw a horizontal line (east-west axis) and a vertical line (north-south axis). The first putt vector will be drawn as a straight line of length 5.0m to the right (east). The second putt vector will be drawn at an angle of 20 degrees from the east, with a length of 2.1m. The third putt vector will be drawn as a straight line of length 0.50m upwards (north).

3. Identify the components of the three vectors:
To solve the problem using components, we need to break down each vector into its x (horizontal) and y (vertical) components.

- The first putt vector (5.0m due east) has only an x-component since it is directed east: Ax = 5.0m, Ay = 0.0m (no vertical component).
- The second putt vector (2.1m at an angle of 20 degrees north of east) can be decomposed into its x and y components. The x-component can be found using the cosine of the angle: Bx = 2.1m * cos(20°), and the y-component can be found using the sine of the angle: By = 2.1m * sin(20°).
- The third putt vector (0.50m due north) has only a y-component since it is directed north: Cx = 0.0m (no horizontal component), Cy = 0.50m.

4. Determine the components of the resultant vector:
To find the resultant vector, we add the x-components and y-components of the three vectors.

Sx = Ax + Bx + Cx
Sy = Ay + By + Cy

Plug in the values obtained for each component.

5. Convert this into the magnitude and direction of the resultant vector:
To find the magnitude and direction of the resultant vector, we use the Pythagorean theorem and trigonometric functions.

Magnitude: IsI = sqrt(Sx^2 + Sy^2)
Angle: Angle = arctan(Sy/Sx)

Evaluate the expressions above using the calculated values of Sx and Sy.

Now, you can follow these steps and perform the calculations to find the magnitude and direction of the resultant vector.