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.