Find the component form of $\mathbf{v}$ given its magnitude and the angle it makes with the positive $x$-axis. Round your answer to four decimals. \[ \|\mathbf{v}\|=9, \theta=150^{\circ} \] Answer
Solution
Step 1 ：We are given the magnitude of the vector \(\|\mathbf{v}\|=9\) and the angle it makes with the positive x-axis \(\theta=150^{\circ}\).
Step 2 ：We can find the component form of the vector using the equations \(v_x = \|v\| \cos(\theta)\) and \(v_y = \|v\| \sin(\theta)\), where \(v_x\) and \(v_y\) are the x and y components of the vector respectively.
Step 3 ：However, the trigonometric functions in the equations expect the angle to be in radians, not degrees. Therefore, we first need to convert the angle from degrees to radians. The conversion factor is \(\pi\) radians = 180 degrees, so \(\theta\) in radians is \(\frac{150}{180}\pi = 2.6179938779914944\).
Step 4 ：Substituting these values into the equations, we find \(v_x = 9 \cos(2.6179938779914944) = -7.7942\) and \(v_y = 9 \sin(2.6179938779914944) = 4.5\).
Step 5 ：Final Answer: The component form of \(\mathbf{v}\) is \(\boxed{(-7.7942, 4.5)}\).
