We are asked to calculate the resultant of three vectors $\vec{a} + \vec{b} + \vec{c}$, where:

1. Vector $\vec{a}$ has a magnitude of 5 and forms an angle of $60^\circ$ with the positive x-axis.
2. Vector $\vec{b} = 5\hat{i} + 8\hat{j}$.
3. Vector $\vec{c}$ has starting and ending points at (4, 5) and (12, 7), respectively.

Step 1: Express each vector in component form

Vector $\vec{a}$

The components of $\vec{a}$ can be derived using trigonometry:

• The x-component is $a_x = 5 \cos(60^\circ)$.
• The y-component is $a_y = 5 \sin(60^\circ)$.

Since $\cos(60^\circ) = \frac{1}{2}$ and $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, we get: $a_x = 5 \times \frac{1}{2} = 2.5$ $a_y = 5 \times \frac{\sqrt{3}}{2} \approx 4.33$ Thus, $\vec{a} = 2.5\hat{i} + 4.33\hat{j}$.

Vector $\vec{b}$

This vector is already given in component form: $\vec{b} = 5\hat{i} + 8\hat{j}$

Vector $\vec{c}$

To find the components of $\vec{c}$, we calculate the differences between the ending and starting points: $c_x = 12 - 4 = 8$ $c_y = 7 - 5 = 2$ Thus, $\vec{c} = 8\hat{i} + 2\hat{j}$.

Step 2: Add the vectors

Now, sum the components of $\vec{a}$, $\vec{b}$, and $\vec{c}$ in the x- and y-directions separately.

x-component:

$R_x = a_x + b_x + c_x = 2.5 + 5 + 8 = 15.5$

y-component:

$R_y = a_y + b_y + c_y = 4.33 + 8 + 2 = 14.33$

Step 3: Calculate the magnitude of the resultant vector

The magnitude $|\vec{R}|$ of the resultant vector is given by the Pythagorean theorem: $|\vec{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{(15.5)^2 + (14.33)^2}$

Let's compute that.

The magnitude of the resultant vector $\vec{R} = \vec{a} + \vec{b} + \vec{c}$ is approximately 21.11.

Would you like more details on any part of the calculation? Here are some questions to consider:

1. How can the direction of the resultant vector be determined?
2. What are unit vectors and how do they affect vector addition?
3. How would the result change if the angle for $\vec{a}$ were different?
4. How can you represent the vectors graphically?
5. What would the calculation look like in 3D?

Tip: Always ensure the angles used in vector calculations are in the correct unit (degrees or radians).