https://r2.erweima.ai/mathbot/image/1725805452610l0c8d5u9.jpg

The problem asks to find the direction of the vector \( \mathbf{v} = (-3, -3) \) in radians.

To determine the direction (or angle) of a vector in the plane, we use the following formula for the angle \( \theta \) of a vector \( \mathbf{v} = (x, y) \):

\[
\theta = \text{atan2}(y, x)
\]

Here, \( x = -3 \) and \( y = -3 \). Now, we compute the angle:

\[
\theta = \text{atan2}(-3, -3)
\]

Since both components are negative, the vector is in the third quadrant. The angle for such a vector is:

\[
\theta = \pi + \text{atan}\left(\frac{3}{3}\right) = \pi + \frac{\pi}{4} = \frac{5\pi}{4}
\]

Thus, the correct answer is option (a) \( \frac{5\pi}{4} \).

Would you like further details or clarification on this solution?

Here are 5 related questions for deeper understanding:
1. How do you find the direction of a vector in the first quadrant?
2. What is the formula for converting an angle in degrees to radians?
3. How do you calculate the magnitude of a vector in two dimensions?
4. Why does the arctangent function result in different values depending on the quadrant?
5. How do you find the unit vector of a given vector?

**Tip:** When dealing with vectors, always pay attention to the signs of the components to correctly determine the quadrant of the direction.

Del vector \( \mathbf{v} = (-3, -3) \), es correcto afirmar que su dirección en radianes es:

Learn how to calculate the direction of the vector \( \mathbf{v} = (-3, -3) \) in radians using trigonometric functions like atan2. This step-by-step explanation is ideal for students in Grades 10-12, focusing on vectors and angle calculations in different quadrants.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation