Questions: Write the vector v in the form a i+b j, where v has the given magnitude and direction v=14, θ=150° v=

Transcript text: Write the vector $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}$, where $\mathbf{v}$ has the given magnitude and direction \[ \begin{array}{c} |v|=14, \theta=150^{\circ} \\ v=\square \end{array} \]

Solution

Solution Steps

To express the vector $\mathbf{v}$ in the form $a \mathbf{i} + b \mathbf{j}$, given its magnitude $|v| = 14$ and direction $\theta = 150^\circ$, we can use trigonometry. The component $a$ is found using $a = |v| \cdot \cos(\theta)$ and the component $b$ is found using $b = |v| \cdot \sin(\theta)$. These calculations will give us the vector in the desired form.

Step 1: Given Information

We are given the magnitude of the vector $|\mathbf{v}| = 14$ and the direction $\theta = 150^\circ$.

Step 2: Convert Angle to Radians

To perform calculations, we convert the angle from degrees to radians: \[ \theta_{\text{radians}} = \frac{150 \cdot \pi}{180} = 2.6179938779914944 \]

Step 3: Calculate the Components

Using the formulas for the components of the vector: \[ a = |\mathbf{v}| \cdot \cos(\theta) = 14 \cdot \cos(2.6179938779914944) \approx -12.1244 \] \[ b = |\mathbf{v}| \cdot \sin(\theta) = 14 \cdot \sin(2.6179938779914944) \approx 7.0000 \]

Step 4: Write the Vector in Component Form

Thus, the vector $\mathbf{v}$ can be expressed in the form: \[ \mathbf{v} = a \mathbf{i} + b \mathbf{j} = -12.1244 \mathbf{i} + 7.0000 \mathbf{j} \]