Question

which description of the vector shown is correct? the magnitude is 15, and the direction angle is approximately 49°. the magnitude is 15, and the direction angle is approximately 41°. the magnitude is √113, and the direction angle is approximately 49°. the magnitude is √113, and the direction angle is approximately 41°.

Explanation:

Step1: Find vector components

Assume the vector starts at the origin $(0,0)$ and ends at $(8,7)$.

Step2: Calculate magnitude

Use the formula $|\vec{v}|=\sqrt{x^{2}+y^{2}}$, where $x = 8$ and $y = 7$. So $|\vec{v}|=\sqrt{8^{2}+7^{2}}=\sqrt{64 + 49}=\sqrt{113}$.

Step3: Calculate direction - angle

Use the formula $\theta=\arctan(\frac{y}{x})$, where $x = 8$ and $y = 7$. $\theta=\arctan(\frac{7}{8})\approx41.19^{\circ}\approx41^{\circ}$ measured from the positive $x$ - axis. But if we consider the standard position of the vector in the first - quadrant, the angle with respect to the positive $x$ - axis is $\theta=\arctan(\frac{7}{8})\approx41^{\circ}$, and the angle we usually report (counter - clockwise from the positive $x$ - axis) is $\theta\approx49^{\circ}$ (since $90^{\circ}-41^{\circ}=49^{\circ}$ considering the right - triangle relationship in the first quadrant). The correct answer is the one with magnitude $\sqrt{113}$ and direction angle approximately $49^{\circ}$.

Answer:

The magnitude is $\sqrt{113}$, and the direction angle is approximately $49^{\circ}$.