Draw each of the following vectors on graph paper, then find the x- and y- components. Make sure to include units in your answers for the components.

a) ⃑r = (65 m, 30degrees above the positive x-axis)
b) ⃑v = (125 m/s, 40degrees below the negative x-axis)
c) ⃑a = (5.3 m/s2, 15degrees to the right of the positive y-axis)

a) Vx = 65 cos 30, Vy = 65 sin 30

b) Vx =125 cos 40, Vy = -125 sin 40

c) Vx = 5.3 sin 30 , Vy = 5.3 cos 30

of course sin 30 = 1/2 and cos 30 = sqrt3/2

Well, I would love to draw these vectors for you, but as a text-based bot, my artistic skills are a bit lacking. However, I can certainly help you with the components!

a) ⃑r = (65 m, 30 degrees above the positive x-axis)
To find the x-component, we can use trigonometry. Since the vector is above the positive x-axis, the y-component will be positive, and the x-component will be the one adjacent to the angle. So, the x-component is:
x = 65 m * cos(30 degrees) ≈ 56.18 m

To find the y-component, we need to look at the side opposite to the angle, which in this case is the y-component itself. So, the y-component is:
y = 65 m * sin(30 degrees) ≈ 32.5 m

b) ⃑v = (125 m/s, 40 degrees below the negative x-axis)
Similarly, to find the x-component, we use trigonometry. Since the vector is below the negative x-axis, both the x- and y-components will be negative. So, the x-component is:
x = -125 m/s * cos(40 degrees) ≈ -95.63 m/s

To find the y-component, we again look at the side opposite to the angle, which is the y-component itself. So, the y-component is:
y = -125 m/s * sin(40 degrees) ≈ -80.19 m/s

c) ⃑a = (5.3 m/s^2, 15 degrees to the right of the positive y-axis)
To find the x-component, we don't need trigonometry since the angle is in respect to the y-axis. So, the x-component is:
x = 0 m/s^2

To find the y-component, we again look at the side opposite to the angle, which is the y-component itself. So, the y-component is:
y = 5.3 m/s^2 * sin(15 degrees) ≈ 1.4 m/s^2

So, the components are:
a) ⃑r = (56.18 m, 32.5 m)
b) ⃑v = (-95.63 m/s, -80.19 m/s)
c) ⃑a = (0 m/s^2, 1.4 m/s^2)

I hope this helps!

To draw each of the vectors on graph paper, we will use the given magnitude and angle to determine the length and direction of the vectors. Let's go step-by-step:

a) ⃑r = (65 m, 30 degrees above the positive x-axis)
1. Start by drawing a coordinate system on the graph paper.
2. Locate the starting point (origin) of the vector ⃑r.
3. From the starting point, draw a line segment in the direction of the angle, which is 30 degrees above the positive x-axis.
4. The length of the line segment should be 65 m. Mark the endpoint of the line segment to represent the vector ⃑r.

Next, let's find the x and y components of ⃑r:
- The x-component is the horizontal displacement of the vector ⃑r. Since the vector is at an angle above the x-axis, the x-component will be positive.
- To find the x-component, we use the formula:
x-component = magnitude * cosine(angle)
x-component = 65 m * cos(30 degrees)
x-component = 65 m * 0.866025 (approximately)
x-component ≈ 56.190975 m

- The y-component is the vertical displacement of the vector ⃑r. Since the vector is at an angle above the x-axis, the y-component will be positive.
- To find the y-component, we use the formula:
y-component = magnitude * sine(angle)
y-component = 65 m * sin(30 degrees)
y-component = 65 m * 0.5
y-component = 32.5 m

So, the x-component of ⃑r is approximately 56.191 m, and the y-component is 32.5 m.

b) ⃑v = (125 m/s, 40 degrees below the negative x-axis)
1. Draw the coordinate system on the graph paper.
2. Locate the starting point (origin) of the vector ⃑v.
3. From the starting point, draw a line segment in the direction of the angle, which is 40 degrees below the negative x-axis.
4. The length of the line segment should be 125 m/s. Mark the endpoint of the line segment to represent the vector ⃑v.

Now, let's find the x and y components of ⃑v:
- The x-component is the horizontal displacement of the vector ⃑v. Since the vector is at an angle below the x-axis, the x-component will be negative.
- To find the x-component, we use the formula:
x-component = magnitude * cosine(angle)
x-component = 125 m/s * cos(40 degrees)
x-component = 125 m/s * 0.766044 (approximately)
x-component ≈ 95.7555 m/s

- The y-component is the vertical displacement of the vector ⃑v. Since the vector is at an angle below the x-axis, the y-component will be negative.
- To find the y-component, we use the formula:
y-component = magnitude * sine(angle)
y-component = 125 m/s * sin(40 degrees)
y-component = 125 m/s * (-0.643068) (approximately)
y-component ≈ -80.3835 m/s

So, the x-component of ⃑v is approximately -95.756 m/s, and the y-component is -80.384 m/s.

c) ⃑a = (5.3 m/s^2, 15 degrees to the right of the positive y-axis)
1. Draw the coordinate system on the graph paper.
2. Locate the starting point (origin) of the vector ⃑a.
3. From the starting point, draw a line segment in the direction of the angle, which is 15 degrees to the right of the positive y-axis.
4. The length of the line segment should be 5.3 m/s^2. Mark the endpoint of the line segment to represent the vector ⃑a.

Finally, let's find the x and y components of ⃑a:
- The x-component is the horizontal displacement of the vector ⃑a. Since the vector is at an angle to the right of the positive y-axis, the x-component will be positive.
- To find the x-component, we use the formula:
x-component = magnitude * cosine(angle)
x-component = 5.3 m/s^2 * cos(15 degrees)
x-component = 5.3 m/s^2 * 0.965926 (approximately)
x-component ≈ 5.119 m/s^2

- The y-component is the vertical displacement of the vector ⃑a. Since the vector is at an angle to the right of the positive y-axis, the y-component will be positive.
- To find the y-component, we use the formula:
y-component = magnitude * sine(angle)
y-component = 5.3 m/s^2 * sin(15 degrees)
y-component = 5.3 m/s^2 * 0.258819 (approximately)
y-component ≈ 1.373 m/s^2

So, the x-component of ⃑a is approximately 5.119 m/s^2, and the y-component is 1.373 m/s^2.

To draw and find the components of each vector, follow these steps:

a) ⃑r = (65 m, 30 degrees above the positive x-axis)

1. Start by drawing a coordinate system on your graph paper, with the positive x-axis to the right and the positive y-axis pointing upwards.
2. From the origin (0,0), draw a line segment 65 units long in the positive x direction to represent the 65 m length.
3. Next, draw a line segment at a 30-degree angle above the positive x-axis, starting from the endpoint of the first line segment.
4. Measure the length of the second line segment drawn to find the component in the y-direction. Multiply this length by sin(30 degrees) to find the y-component of the vector.
y-component = length * sin(30 degrees)
5. Measure the length of the second line segment drawn to find the component in the x-direction. Multiply this length by cos(30 degrees) to find the x-component of the vector.
x-component = length * cos(30 degrees)

b) ⃑v = (125 m/s, 40 degrees below the negative x-axis)

1. Draw a coordinate system on your graph paper, similar to step 1 above, but with the negative x-axis going to the left and the positive y-axis pointing upwards.
2. From the origin (0,0), draw a line segment 125 units long to the left (negative x direction) to represent the 125 m/s length.
3. Next, draw a line segment at a 40-degree angle below the negative x-axis, starting from the endpoint of the first line segment.
4. Measure the length of the second line segment drawn to find the component in the y-direction. Multiply this length by sin(40 degrees) to find the y-component of the vector.
y-component = length * sin(40 degrees)
5. Measure the length of the second line segment drawn to find the component in the x-direction. Multiply this length by cos(40 degrees) to find the x-component of the vector.
x-component = length * cos(40 degrees)

c) ⃑a = (5.3 m/s^2, 15 degrees to the right of the positive y-axis)

1. Draw a coordinate system on your graph paper, similar to step 1 above.
2. From the origin (0,0), draw a line segment 5.3 units long in the positive y direction to represent the 5.3 m/s^2 length.
3. Next, draw a line segment at a 15-degree angle to the right of the positive y-axis, starting from the endpoint of the first line segment.
4. Measure the length of the second line segment drawn to find the component in the y-direction. Multiply this length by cos(15 degrees) to find the y-component of the vector.
y-component = length * cos(15 degrees)
5. Measure the length of the second line segment drawn to find the component in the x-direction. Multiply this length by sin(15 degrees) to find the x-component of the vector.
x-component = length * sin(15 degrees)

Remember to label the x- and y-components for each vector with their respective units (meters, meters per second, or meters per second squared).