The magnitudes of the four displacement vectors shown in the drawing are A = 15.0 m, B = 11.0 m, C = 12.0 m, and D = 21.0 m. Determine the magnitude and directional angle for the resultant that occurs when these vectors are added together.

What are the angles?

To determine the magnitude and directional angle of the resultant vector, we need to find the sum of the four displacement vectors.

1. Start by drawing a diagram to represent the vectors. Label vectors A, B, C, and D with their corresponding magnitudes.

A

C ← ─┼─→ D

B

2. To find the sum of the displacement vectors, we'll use the Pythagorean theorem to find the magnitude. The magnitude of the resultant vector R is given by:

R = √(A^2 + B^2 + C^2 + D^2)

Substituting the values given:

R = √(15.0^2 + 11.0^2 + 12.0^2 + 21.0^2)
= √(225 + 121 + 144 + 441)
= √(931)

Therefore, the magnitude of the resultant vector R is approximately 30.5 m.

3. To determine the directional angle, we'll use the trigonometric functions sine and cosine.

Let's break down the vectors into their x and y components as follows:

A = Ax î + Ay ĵ
B = Bx î + By ĵ
C = Cx î + Cy ĵ
D = Dx î + Dy ĵ

Now, we can find the x and y components of the resultant vector R:

Rx = Ax + Bx + Cx + Dx
Ry = Ay + By + Cy + Dy

The directional angle θ can be calculated using the arctangent function:

θ = tan^(-1)(Ry / Rx)

4. Let's calculate the x and y components of the resultant vector:

Rx = 0 + 0 + 0 + 0 = 0
Ry = 0 + 0 + 0 + 0 = 0

Since the x and y components are both zero, we can conclude that the magnitude and direction of the resultant vector are both zero.

Therefore, the magnitude of the resultant vector is 0 m and the directional angle is 0 degrees.