To write the vector in component form using its magnitude and direction, you can apply the following formulas:

$v_x = |v| \cdot \cos(\theta)$ $v_y = |v| \cdot \sin(\theta)$

Where:

• $v_x$ is the horizontal component,
• $v_y$ is the vertical component,
• $|v|$ is the magnitude of the vector,
• $\theta$ is the direction (angle) in standard position.

Given:

• $|v| = 6$,
• $\theta = 45^\circ$.

First, calculate $v_x$ and $v_y$:

$v_x = 6 \cdot \cos(45^\circ) = 6 \cdot \frac{\sqrt{2}}{2} = 6 \cdot 0.7071 \approx 4.2426$

$v_y = 6 \cdot \sin(45^\circ) = 6 \cdot \frac{\sqrt{2}}{2} = 6 \cdot 0.7071 \approx 4.2426$

Thus, the vector in component form is approximately:

$\mathbf{v} = (4.24, 4.24)$

Would you like more details or have any questions?

Here are some related questions:

1. How do you convert between polar and rectangular coordinates?
2. What is the formula to find the magnitude of a vector from its components?
3. How do you find the angle of a vector if its components are given?
4. What is the significance of unit vectors and how are they calculated?
5. How can you find the direction of a vector in degrees from its components?

Tip: For angles like $45^\circ$, using the symmetry of trigonometric values like $\frac{\sqrt{2}}{2}$ makes calculations easier!