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3.2.1. Decompose a 2D vector trigonometrically

Give the formulas for the x and y

components of the vector 𝐴𝐵$$$$$⃗ when the

angle 𝛼 with the positive x-axis and the

length l are given.

To decompose the 2D vector $ \vec{AB} $ trigonometrically into its $ x $- and $ y $-components, we use the following approach:

### Given:
- $ l $: the magnitude (length) of the vector $ \vec{AB} $
- $ \alpha $: the angle the vector makes with the positive $ x $-axis.

### Formulas for components:
- **$ x $-component**: $ AB_x = l \cdot \cos(\alpha) $
- **$ y $-component**: $ AB_y = l \cdot \sin(\alpha) $

Here:
- $ l $ is the magnitude of the vector $ \vec{AB} $.
- $ \alpha $ is the angle of the vector with respect to the positive direction of the $ x $-axis.
- The cosine and sine functions give us the respective components along the $ x $- and $ y $-axes.

Would you like further details or clarifications on this topic?

### 5 related questions:
1. How do we calculate the magnitude of a vector if only the components are given?
2. What happens to the components when $ \alpha $ is greater than $ 90^\circ $?
3. Can we derive these component formulas using unit vectors?
4. How does the direction of the vector affect its $ x $- and $ y $-components?
5. What is the relationship between the angle $ \alpha $ and the direction of the vector?

### Tip:
Always make sure the angle $ \alpha $ is measured from the positive $ x $-axis and in the correct direction (usually counterclockwise) to avoid sign errors in the components.

Decompose a 2D vector trigonometrically. Give the formulas for the x and y components of the vector AB⃗ when the angle α with the positive x-axis and the length l are given.

Learn how to decompose a 2D vector into its x and y components using trigonometric functions. This problem involves calculating the vector's components when the angle α and length l are given, using cosine and sine formulas. Suitable for high school students in Grades 9-12.

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