Question

describing a vector
which description of the vector shown is correct?
the magnitude is 7√2, and the direction angle is approximately 315°.
the magnitude is 7√2, and the direction angle is approximately 135°.
the magnitude is √6, and the direction angle is approximately 315°.
the magnitude is √6, and the direction angle is approximately 135°.

Explanation:

Step1: Identify vector components

Let the initial - point of the vector be at the origin $(0,0)$ and the terminal - point be $(- 7,7)$.

Step2: Calculate the magnitude

Use the magnitude formula $|\vec{v}|=\sqrt{x^{2}+y^{2}}$. Here, $x=-7$ and $y = 7$. So $|\vec{v}|=\sqrt{(-7)^{2}+7^{2}}=\sqrt{49 + 49}=\sqrt{98}=7\sqrt{2}$.

Step3: Calculate the direction angle

Use the formula $\theta=\arctan(\frac{y}{x})$. Here, $\theta=\arctan(\frac{7}{-7})=\arctan(-1)$. Since the vector lies in the second quadrant (because $x\lt0$ and $y\gt0$), $\theta = 135^{\circ}$.

Answer:

The magnitude is $7\sqrt{2}$, and the direction angle is approximately $135^{\circ}$.