Let's break this problem down step by step, starting with calculating each vector in terms of its components.

Step 1: Vector $\mathbf{a}$

• The magnitude of $\mathbf{a}$ is given as 5, and it forms an angle of $60^\circ$ with the positive x-axis.
• Using trigonometry, we can find the components of $\mathbf{a}$ along the $\hat{x}$ and $\hat{y}$ directions.

$\mathbf{a_x} = 5 \cdot \cos(60^\circ) = 5 \cdot \frac{1}{2} = 2.5$ $\mathbf{a_y} = 5 \cdot \sin(60^\circ) = 5 \cdot \frac{\sqrt{3}}{2} \approx 4.33$ So, vector $\mathbf{a}$ can be expressed as: $\mathbf{a} = 2.5 \hat{x} + 4.33 \hat{y}$

Step 2: Vector $\mathbf{b}$

• Vector $\mathbf{b}$ is directly given as $\mathbf{b} = 5 \hat{x} + 8 \hat{y}$.

Step 3: Vector $\mathbf{c}$

• The coordinates of the starting point of vector $\mathbf{c}$ are $(4, 5)$ and the ending point is $(12, 7)$.
• To find $\mathbf{c}$, we calculate the difference between the ending and starting coordinates.

$\mathbf{c_x} = 12 - 4 = 8$ $\mathbf{c_y} = 7 - 5 = 2$ Thus, vector $\mathbf{c}$ can be expressed as: $\mathbf{c} = 8 \hat{x} + 2 \hat{y}$

Step 4: Sum of the vectors $\mathbf{a} + \mathbf{b} + \mathbf{c}$

Now, let's sum the three vectors component-wise.

x-component:

$\mathbf{R_x} = \mathbf{a_x} + \mathbf{b_x} + \mathbf{c_x} = 2.5 + 5 + 8 = 15.5$

y-component:

$\mathbf{R_y} = \mathbf{a_y} + \mathbf{b_y} + \mathbf{c_y} = 4.33 + 8 + 2 = 14.33$

Step 5: Magnitude of the resultant vector

The magnitude $R$ of the resultant vector can be found using the Pythagorean theorem: $R = \sqrt{(\mathbf{R_x})^2 + (\mathbf{R_y})^2} = \sqrt{(15.5)^2 + (14.33)^2}$ $R \approx \sqrt{240.25 + 205.35} = \sqrt{445.6} \approx 21.11$

Step 6: Direction of the resultant vector

The direction $\theta$ (with respect to the positive x-axis) can be found using the tangent function: $\theta = \tan^{-1} \left( \frac{\mathbf{R_y}}{\mathbf{R_x}} \right) = \tan^{-1} \left( \frac{14.33}{15.5} \right)$ $\theta \approx \tan^{-1} (0.924) \approx 42.67^\circ$

Final Result:

The resultant vector $\mathbf{R}$ has:

• Magnitude: approximately $21.11$
• Direction: approximately $42.67^\circ$ above the positive x-axis.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the result change if the angle of vector $\mathbf{a}$ was different?
2. Can you verify the magnitude and direction of the resultant vector using graphical methods?
3. How does the position of vector $\mathbf{c}$ affect the overall resultant?
4. What would happen if vector $\mathbf{b}$ was negative in both components?
5. How can you decompose the resultant vector into unit vector form?

Tip: Always keep track of the angle direction (above or below the x-axis) when calculating vector components!