Question

1) Compute the x and y components of the following vectors, and state them in component form.

a) A 8.0 m South
b) B-15.0 m at 30-East of North
c) C = 12.0 m at 25-South of West -
d) D=10.0 m at 53-West of North-

Answer 1

a) A 8.0 m South:
The x component of vector A is 0 since it points purely in the y direction (South), while the y component is -8.0 m (negative since it points downwards):
A = (0, -8.0 m)

b) B -15.0 m at 30-East of North:
To find the components, we first visualize the vector as shown below:

N
|
|
| 30°
| /
|/
+------------- E


The x component of B is found by projecting the vector onto the x-axis (which is East). This gives us:
Bx = -15.0 m * sin(30°) = -7.5 m

The y component of B is found by projecting the vector onto the y-axis (which is North). This gives us:
By = -15.0 m * cos(30°) = -13.0 m

Therefore, the component form of vector B is:
B = (-7.5 m, -13.0 m)

c) C = 12.0 m at 25-South of West:
To find the components, we first visualize the vector as shown below:

N
|
|
|
|
| 25°
| /
|/
+------------- W

The x component of C is found by projecting the vector onto the x-axis (which is West). This gives us:
Cx = -12.0 m * cos(25°) = -10.9 m

The y component of C is found by projecting the vector onto the y-axis (which is North). This gives us:
Cy = -12.0 m * sin(25°) = -5.1 m

Therefore, the component form of vector C is:
C = (-10.9 m, -5.1 m)

d) D = 10.0 m at 53-West of North:
To find the components, we first visualize the vector as shown below:

N
|
|
| 53°
| /
|/
+------------- W

The x component of D is found by projecting the vector onto the x-axis (which is West). This gives us:
Dx = -10.0 m * sin(53°) = -8.1 m

The y component of D is found by projecting the vector onto the y-axis (which is North). This gives us:
Dy = 10.0 m * cos(53°) = 6.2 m

Therefore, the component form of vector D is:
D = (-8.1 m, 6.2 m)

Answer 2

a) A = 8.0 m South

Since the vector is directly along the South direction, there is no x component.

x component: 0 m

y component: -8.0 m (negative because it's southward)

Component form: A = (0, -8.0)

b) B = -15.0 m at 30° East of North

To find the components, we can use the following relationships:

x component: B_x = B * sin(θ)

y component: B_y = B * cos(θ)

B_x = -15.0 * sin(30°) = -15.0 * 0.5 = -7.5 m

B_y = -15.0 * cos(30°) = -15.0 * (sqrt(3)/2) ≈ -12.99 m

Component form: B ≈ (-7.5, -12.99)

c) C = 12.0 m at 25° South of West

x component: C_x = -C * cos(θ) (negative because it's westward)

y component: C_y = -C * sin(θ) (negative because it's southward)

C_x = -12.0 * cos(25°) ≈ -10.85 m

C_y = -12.0 * sin(25°) ≈ -5.16 m

Component form: C ≈ (-10.85, -5.16)

d) D = 10.0 m at 53° West of North

x component: D_x = -D * sin(θ) (negative because it's westward)

y component: D_y = D * cos(θ)

D_x = -10.0 * sin(53°) ≈ -8.0 m

D_y = 10.0 * cos(53°) ≈ 6.0 m

Component form: D ≈ (-8.0, 6.0)