Questions: Write the vector v in the form al+bj, given its magnitude v and the angle α it makes with the positive x-axis. [ v=7, quad alpha=60^circ ] [ mathrmv=square ] (Simplify your answer. Type an exact answer using radicals as needed. Type your answer in the form al + bj.)

Solution Steps

To write the vector \( \mathbf{v} \) in the form \( a\mathbf{i} + b\mathbf{j} \), given its magnitude \( |\mathbf{v}| \) and the angle \( \alpha \) it makes with the positive x-axis, we can use the following steps:

1. Calculate the x-component \( a \) of the vector using \( a = |\mathbf{v}| \cos(\alpha) \).
2. Calculate the y-component \( b \) of the vector using \( b = |\mathbf{v}| \sin(\alpha) \).
3. Combine these components to express the vector in the form \( a\mathbf{i} + b\mathbf{j} \).

To work with trigonometric functions, we first convert the angle \( \alpha = 60^\circ \) to radians: \[ \alpha_{\text{radians}} = \frac{\pi}{3} \approx 1.0472 \]

Using the formula for the x-component of the vector: \[ a = |\mathbf{v}| \cos(\alpha) = 7 \cos\left(\frac{\pi}{3}\right) = 7 \cdot \frac{1}{2} = 3.5 \]

Using the formula for the y-component of the vector: \[ b = |\mathbf{v}| \sin(\alpha) = 7 \sin\left(\frac{\pi}{3}\right) = 7 \cdot \frac{\sqrt{3}}{2} \approx 6.0622 \]

Combining the components, we express the vector \( \mathbf{v} \) in the form \( a\mathbf{i} + b\mathbf{j} \): \[ \mathbf{v} = 3.5\mathbf{i} + 6.0622\mathbf{j} \]

Final Answer