Resolving Vectors into Components

Level 1 - Physics topic page in Measurement and Vectors.

Principle

Components replace one vector with perpendicular signed projections.

\(\vec{a}\)

vector

varies

\(a\)

magnitude of vector

same as vector

\(a_x\)

x-component

same as vector

\(a_y\)

y-component

same as vector

\(\theta\)

angle from positive x-axis

rad or deg

The diagram shows components as projections onto fixed perpendicular axes.

The horizontal and vertical components reconstruct the original vector.

Projection gives signed component lengths; reconstruction adds those perpendicular pieces back to the original vector.

Horizontal projection

\[a_x=a\cos\theta\]

Vertical projection

\[a_y=a\sin\theta\]

Reconstruct

\[\vec{a}=a_x\hat{\imath}+a_y\hat{\jmath}\]

Magnitude

\[a=\sqrt{a_x^2+a_y^2}\]

Project components

\[a_x=a\cos\theta,\quad a_y=a\sin\theta\]

Rebuild vector

\[\vec{a}=a_x\hat{\imath}+a_y\hat{\jmath}\]

Magnitude check

\[a=\sqrt{a_x^2+a_y^2}\]

Direction check

\[\tan\theta=\frac{a_y}{a_x}\]