To find the vector $v$ in component form with a given magnitude and direction, we can use the following formulas for the components of a vector given its magnitude $|v|$ and direction angle $\theta$:

$v_x = |v| \cos \theta$ $v_y = |v| \sin \theta$

Given:

• Magnitude $|v| = \frac{1}{4}$
• Direction angle $\theta = 30^\circ$

Let's calculate each component.

1. Calculate $v_x$: $v_x = \frac{1}{4} \cos 30^\circ = \frac{1}{4} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8}$

2. Calculate $v_y$: $v_y = \frac{1}{4} \sin 30^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$

Thus, the vector $v$ in component form is: $v = \left( \frac{\sqrt{3}}{8}, \frac{1}{8} \right)$

Would you like further explanation on any step, or do you have additional questions?

Here are some related questions you might find useful:

1. How do we convert a vector from component form to magnitude and direction?
2. What are the trigonometric values of common angles (like 30°, 45°, 60°)?
3. How can we determine the direction angle of a vector given its components?
4. How do vectors in different quadrants affect the sign of their components?
5. What is the importance of unit vectors in vector operations?

Tip: For any vector in component form, you can always verify the magnitude by applying the Pythagorean theorem to its components.