Write the vector \( v \) in the form a \( +b \mathrm{j} \), given its magnitude \( \|v\| \) and the angle \( \alpha \) it makes with the positive \( x \)-axis. \[ \|v\|=5 \text {, } \]

To express the vector \( v \) in the form \( a + b \mathrm{j} \), we can use the magnitude \( \|v\| \) and the angle \( \alpha \) it makes with the positive \( x \)-axis. The components of the vector can be calculated using the following formulas: \[ a = \|v\| \cos(\alpha) \] \[ b = \|v\| \sin(\alpha) \] Given that \( \|v\| = 5 \), we can substitute this value into the equations: \[ a = 5 \cos(\alpha) \] \[ b = 5 \sin(\alpha) \] Thus, the vector \( v \) can be expressed as: \[ v = a + b \mathrm{j} = 5 \cos(\alpha) + 5 \sin(\alpha) \mathrm{j} \] So, the vector \( v \) in the form \( a + b \mathrm{j} \) is: \[ v = 5 \cos(\alpha) + 5 \sin(\alpha) \mathrm{j} \]