Vector Decomposition and Direction

In physics, vectors represent quantities that have both magnitude and direction, such as velocity, acceleration, and force.

Vector Decomposition

To analyze motion in two dimensions, we often break a vector into components along the x and y axes. This is called vector decomposition.

For a vector \( \vec{A} \) with magnitude \( A \) and angle \( \theta \) from the horizontal:

\( A_x = A \cos(\theta) \)
\( A_y = A \sin(\theta) \)

where:

\( A_x \) is the horizontal component
\( A_y \) is the vertical component

Visualizing Vector Decomposition

Angle (°): 30

Magnitude: 60

Vector \( \vec{A} \): Blue arrow

Horizontal component \( A_x = A \cos \theta \): Red arrow

Vertical component \( A_y = A \sin \theta \): Green arrow

Angle \( \theta \): Between horizontal axis and \( \vec{A} \)

Reconstructing Magnitude and Angle

You can reconstruct the magnitude and direction of a vector if you know its components. For example, using velocity components \( v_x \) and \( v_y \):

\[ v = \sqrt{v_x^2 + v_y^2} \]
\[ \theta = \tan^{-1}\left(\frac{v_y}{v_x}\right) \]

This same process can be applied to any vector (force, acceleration, displacement, etc.) by replacing the velocity components with the appropriate vector components. This is a very important piece of 2D motion because for many questions, you will first have to break velocity vectors into components and solve for individual components. Later on, the question can ask you to find the magnitude of velocity and the direction, which can be done when finding the component magnitudes