Question

find the magnitude and direction angle (in degrees) of the following vector.
v = 7(cos(80°)i + sin(80°)j)

Explanation:

Step1: Recall the formula for the magnitude of a vector in polar form

For a vector \(\mathbf{v}=r(\cos\theta\mathbf{i}+\sin\theta\mathbf{j})\), the magnitude \(||\mathbf{v}|| = r\).

Step2: Identify the magnitude

Given \(\mathbf{v}=7(\cos(80^{\circ})\mathbf{i}+\sin(80^{\circ})\mathbf{j})\), by comparing with \(\mathbf{v}=r(\cos\theta\mathbf{i}+\sin\theta\mathbf{j})\), we have \(r = 7\). So the magnitude \(||\mathbf{v}||=7\).

Step3: Recall the formula for the direction angle

For a vector \(\mathbf{v}=r(\cos\theta\mathbf{i}+\sin\theta\mathbf{j})\), the direction angle \(\alpha=\theta\).

Step4: Identify the direction angle

Given \(\mathbf{v}=7(\cos(80^{\circ})\mathbf{i}+\sin(80^{\circ})\mathbf{j})\), by comparing with \(\mathbf{v}=r(\cos\theta\mathbf{i}+\sin\theta\mathbf{j})\), we have \(\theta = 80^{\circ}\). So the direction angle \(\alpha = 80^{\circ}\).

Answer:

The magnitude of the vector \(\mathbf{v}\) is \(7\) and the direction angle is \(80^{\circ}\).