What is the representation of vector l using magnitude and direction?

Vector \( \mathbf{l} \) is represented by its magnitude and the angle it makes with a reference direction.

In more detail, a vector is a quantity that has both magnitude (size) and direction. To represent vector \( \mathbf{l} \) using its magnitude and direction, you need two pieces of information: the length of the vector (its magnitude) and the angle it forms with a reference direction, usually the positive x-axis in a coordinate system.

The magnitude of vector \( \mathbf{l} \) is often denoted as \( |\mathbf{l}| \) and can be found using the Pythagorean theorem if you know its components. For example, if vector \( \mathbf{l} \) has components \( l_x \) and \( l_y \) in the x and y directions respectively, its magnitude is calculated as \( |\mathbf{l}| = \sqrt{l_x^2 + l_y^2} \).

The direction of vector \( \mathbf{l} \) is given by the angle \( \theta \) it makes with the positive x-axis. This angle can be found using trigonometry, specifically the tangent function: \( \theta = \tan^{-1} \left( \frac{l_y}{l_x} \right) \). This angle is usually measured in degrees or radians.

So, if you know the magnitude \( |\mathbf{l}| \) and the angle \( \theta \), you can fully describe the vector \( \mathbf{l} \). This representation is very useful in physics and engineering, where understanding both the size and direction of a vector is crucial.