find the magnitude of the vector sum -->a +-->b.

Question

vector a is 3250m long in a 38.4 degrees direction.
vector b is 1822m long in a 58.8 direction.

Answer 1

To find the magnitude of the vector sum \(\mathbf{a} + \mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are given vectors, we follow these steps:

1. Decompose each vector into its components.
2. Add the respective components to get the resultant vector.
3. Calculate the magnitude of the resultant vector.

Given:
- \(\mathbf{a}\) has a magnitude of 3250 m and direction of 38.4 degrees.
- \(\mathbf{b}\) has a magnitude of 1822 m and direction of 58.8 degrees.

### Step 1: Find the components of each vector

For \(\mathbf{a}\):
- \( \mathbf{a_x} = 3250 \cos(38.4^\circ) \)
- \( \mathbf{a_y} = 3250 \sin(38.4^\circ) \)

For \(\mathbf{b}\):
- \( \mathbf{b_x} = 1822 \cos(58.8^\circ) \)
- \( \mathbf{b_y} = 1822 \sin(58.8^\circ) \)

First compute the cosine and sine values:
- \( \cos(38.4^\circ) \approx 0.785 \)
- \( \sin(38.4^\circ) \approx 0.620 \)

- \( \cos(58.8^\circ) \approx 0.517 \)
- \( \sin(58.8^\circ) \approx 0.856 \)

Calculating the components:
\[ \mathbf{a_x} \approx 3250 \times 0.785 = 2551.25 \, \text{m} \]
\[ \mathbf{a_y} \approx 3250 \times 0.620 = 2015.00 \, \text{m} \]

\[ \mathbf{b_x} \approx 1822 \times 0.517 = 941.974 \, \text{m} \]
\[ \mathbf{b_y} \approx 1822 \times 0.856 = 1559.632 \, \text{m} \]

### Step 2: Add the components

Combine the x-components and y-components to get the components of the resultant vector \(\mathbf{R}\):
\[ \mathbf{R_x} = \mathbf{a_x} + \mathbf{b_x} = 2551.25 + 941.974 = 3493.224 \, \text{m} \]
\[ \mathbf{R_y} = \mathbf{a_y} + \mathbf{b_y} = 2015.00 + 1559.632 = 3574.632 \, \text{m} \]

### Step 3: Calculate the magnitude of the resultant vector

The magnitude \( R \) can be found using the Pythagorean theorem:
\[ R = \sqrt{\mathbf{R_x}^2 + \mathbf{R_y}^2} \]

Plugging in the values:
\[ R = \sqrt{(3493.224)^2 + (3574.632)^2} \]

\[ R \approx \sqrt{12152393.8 + 12792656.2} \]
\[ R \approx \sqrt{24945050} \]
\[ R \approx 4994.50 \, \text{m} \]

So, the magnitude of the vector sum \(\mathbf{a} + \mathbf{b}\) is approximately \( 4994.50 \, \text{m} \).