Question

A vector has components (3, -4). What is the angle it makes with the positive x-axis?

• A. 60
• B. 53.13
• C. 90
• D. 45

Answer 1

Answer: (B)

53.13

Answer 2

To find the angle a vector makes with the positive x-axis, we can use its components. Let the vector be $\vec{V} = (V_x, V_y) = (3, -4)$. Here, $V_x = 3$ and $V_y = -4$.

The angle $\theta$ that the vector makes with the positive x-axis can be found using the tangent function: $\tan \theta = \frac{V_y}{V_x}$ $\tan \theta = \frac{-4}{3}$

Since $V_x$ is positive (3) and $V_y$ is negative (-4), the vector lies in the fourth quadrant. The value of $\tan \theta$ is negative, which is consistent with an angle in the fourth quadrant.

To find the reference angle (the acute angle the vector makes with the x-axis), let's denote it as $\alpha$. $\tan \alpha = \left| \frac{V_y}{V_x} \right| = \left| \frac{-4}{3} \right| = \frac{4}{3}$ So, $\alpha = \arctan\left(\frac{4}{3}\right)$.

Using a calculator, $\arctan(4/3) \approx 53.13^\circ$.

Now, let's consider the actual angle $\theta$ with the positive x-axis. Since the vector is in the fourth quadrant, the angle $\theta$ can be expressed as: $\theta = 360^\circ - \alpha$ (measured counter-clockwise from the positive x-axis) $\theta = 360^\circ - 53.13^\circ = 306.87^\circ$ OR $\theta = -\alpha$ (measured clockwise from the positive x-axis) $\theta = -53.13^\circ$

Looking at the given options: A) 60 B) 53.13 C) 90 D) 45

Neither $306.87^\circ$ nor $-53.13^\circ$ are among the options. However, $53.13^\circ$ is one of the options. In multiple-choice questions of this type, when the actual angle (in standard position) is not an option, but the reference angle (the acute angle with the x-axis) is, the reference angle is often the intended answer. The question might implicitly be asking for the magnitude of the acute angle formed by the vector with the x-axis.

Therefore, the angle is approximately $53.13^\circ$.