Using the given magnitude and direction in standard position, write the vector in component form. |v| = 6, 𝜃 = 45°

To write the vector in component form using its magnitude and direction, you can apply the following formulas:

\[
v_x = |v| \cdot \cos(\theta)
\]
\[
v_y = |v| \cdot \sin(\theta)
\]

Where:
- \(v_x\) is the horizontal component,
- \(v_y\) is the vertical component,
- \(|v|\) is the magnitude of the vector,
- \(\theta\) is the direction (angle) in standard position.

Given:
- \(|v| = 6\),
- \(\theta = 45^\circ\).

First, calculate \(v_x\) and \(v_y\):

\[
v_x = 6 \cdot \cos(45^\circ) = 6 \cdot \frac{\sqrt{2}}{2} = 6 \cdot 0.7071 \approx 4.2426
\]

\[
v_y = 6 \cdot \sin(45^\circ) = 6 \cdot \frac{\sqrt{2}}{2} = 6 \cdot 0.7071 \approx 4.2426
\]

Thus, the vector in component form is approximately:

\[
\mathbf{v} = (4.24, 4.24)
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you convert between polar and rectangular coordinates?
2. What is the formula to find the magnitude of a vector from its components?
3. How do you find the angle of a vector if its components are given?
4. What is the significance of unit vectors and how are they calculated?
5. How can you find the direction of a vector in degrees from its components?

**Tip:** For angles like \(45^\circ\), using the symmetry of trigonometric values like \(\frac{\sqrt{2}}{2}\) makes calculations easier!

Learn how to express a vector in component form using a magnitude of 6 and an angle of 45°. This example covers vector decomposition into horizontal and vertical components using trigonometry. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation