Find the component form of \( \mathbf{v} \) given its magnitude and the angle it makes with the positive \( x \)-axis. \[ \begin{array}{l}\text { magnitude } \\ \|\mathbf{v}\|=8 \quad \mathbf{v} \text { in the direction } 3 \mathbf{i}+4 \mathbf{j}\end{array} \]

To find the component form of the vector \( \mathbf{v} \), we start with its magnitude and the direction it is pointing. Given that \( \|\mathbf{v}\| = 8 \) and the direction is represented by the vector \( 3 \mathbf{i} + 4 \mathbf{j} \), we first need to determine the unit vector in the direction of \( 3 \mathbf{i} + 4 \mathbf{j} \). The magnitude of the direction vector \( 3 \mathbf{i} + 4 \mathbf{j} \) is calculated as follows: \[ \|\mathbf{d}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] Next, we find the unit vector \( \mathbf{u} \) in the direction of \( 3 \mathbf{i} + 4 \mathbf{j} \): \[ \mathbf{u} = \frac{1}{\|\mathbf{d}\|} (3 \mathbf{i} + 4 \mathbf{j}) = \frac{1}{5} (3 \mathbf{i} + 4 \mathbf{j}) = \frac{3}{5} \mathbf{i} + \frac{4}{5} \mathbf{j}. \] Now, to find the component form of \( \mathbf{v} \), we multiply the unit vector \( \mathbf{u} \) by the magnitude \( \|\mathbf{v}\| = 8 \): \[ \mathbf{v} = \|\mathbf{v}\| \cdot \mathbf{u} = 8 \left( \frac{3}{5} \mathbf{i} + \frac{4}{5} \mathbf{j} \right) = \frac{24}{5} \mathbf{i} + \frac{32}{5} \mathbf{j}. \] Thus, the component form of the vector \( \mathbf{v} \) is \[ \mathbf{v} = \left( \frac{24}{5}, \frac{32}{5} \right). \]