To write the vector $\mathbf{v}$ in the form $a\mathbf{i} + b\mathbf{j}$, we use the following relationships based on the magnitude and direction of the vector:

$a = ||\mathbf{v}|| \cos \alpha$ $b = ||\mathbf{v}|| \sin \alpha$

Given:

$||\mathbf{v}|| = 3, \quad \alpha = 30^\circ$

Step 1: Compute $a$ and $b$

$a = 3 \cos 30^\circ = 3 \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$ $b = 3 \sin 30^\circ = 3 \times \frac{1}{2} = \frac{3}{2}$

Step 2: Write the vector in component form

$\mathbf{v} = \frac{3\sqrt{3}}{2} \mathbf{i} + \frac{3}{2} \mathbf{j}$

Thus, the vector in the form $a\mathbf{i} + b\mathbf{j}$ is:

$\mathbf{v} = \frac{3\sqrt{3}}{2} \mathbf{i} + \frac{3}{2} \mathbf{j}$

Would you like me to further explain or assist with related concepts?

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Related Questions:

1. How do you compute the magnitude of a vector given its components?
2. What are the unit vector equivalents for $\mathbf{i}$ and $\mathbf{j}$?
3. How is the angle of a vector measured relative to the x-axis?
4. How do trigonometric functions relate to vectors in 2D space?
5. Can this representation be extended to 3D vectors? How?

Tip: Always confirm your calculator is set to the correct mode (degrees/radians) when working with angles.