Vectors and Components

Decomposition of vectors into Cartesian components and calculation of magnitude and angle.

Conceptual context

When an object moves, the force driving it or the velocity it carries rarely points exactly along a single axis. To describe that direction precisely and to operate algebraically with it, physics uses vectors: mathematical objects that have both a magnitude and a direction. Decomposing a vector into its Cartesian components is equivalent to asking: how much of this vector points to the right, and how much points upward? The answer, expressed as signed numbers, turns a two-dimensional geometric problem into two independent scalar problems that can be solved separately. That is the central power of this leaf.

🟢 Essential level

A vector in the plane is an arrow with length $v$ and direction $\theta$, measured from the positive horizontal axis counterclockwise.

Decomposition asks: what shadow does that arrow cast on the horizontal axis, and what shadow on the vertical axis? The horizontal shadow is component $v_x$ and the vertical shadow is component $v_y$.

When the vector points to the right, all its length is in $v_x$ and $v_y$ is zero. When it points straight up, all the length is in $v_y$ and $v_x$ is zero. For any other direction the length is shared between both components according to the cosine and sine of $\theta$. Neither component can exceed the magnitude $v$ in absolute value; the total length is always conserved through the Pythagorean relation.

For example, a vector of 10 units pointing 60° above the horizontal splits its length unevenly between the two axes: most of it ends up in the vertical component, because the angle is closer to 90° than to 0°, leaving only a smaller fraction in the horizontal component.

🔵 Formal level

Basic formulas

The four core formulas of this leaf connect the magnitude $v$, angle $\theta$, and components $v_x$, $v_y$ in both directions.

x-component:

x-component of a vector

$v_x$ is the projection of the vector onto the x-axis. The cosine equals 1 when $\theta$ = 0 (horizontal vector) and 0 when $\theta$ = π/2 (vertical vector). The sign of $v_x$ indicates which horizontal direction the projection faces.

y-component:

y-component of a vector

$v_y$ is the projection onto the y-axis. The sine equals 0 when $\theta$ = 0 and reaches its maximum when $\theta$ = π/2. Both formulas always work as a pair: increasing $\theta$ from 0 to π/2 transfers length from $v_x$ to $v_y$.

Real student questions

Why can $v_x$ be negative if $v$ is positive? $v$ measures the total length of the vector, which cannot be negative. $v_x$ measures the projection onto the x-axis with a sign: if the vector points to the left relative to the x-axis, the projection is negative. The magnitude is always non-negative; the components are signed quantities.

When should I use atan and when atan2? Always use atan2($v_y$, $v_x$) to compute $\theta$. The function atan($v_y$/$v_x$) gives the correct angle only when $v_x$ > 0. In the second quadrant ($v_x$ < 0, $v_y$ > 0), atan returns an incorrect negative angle; atan2 returns the correct angle in (−π, π].

Why must $\theta$ be in radians? Scientific calculators and programming languages compute cos and sin in radians by default. cos(π/4) is equal to 0.707, but cos(45) ≈ 0.526 (in radians, 45 rad ≈ 14.3 full turns). Using degrees without converting gives a completely wrong result.

Is the magnitude the same as the sum of the components? No. The magnitude is $v$ is equal to sqrt($v_x$² + $v_y$²), which is always less than or equal to |$v_x$| + |$v_y$| (triangle inequality). For $\theta$ is equal to π/4, $v_x$ + $v_y$ is equal to 1.414$v$, but $v$ is equal to 1$v$.

Final synthesis

Decomposing a vector means expressing its direction and magnitude in the language of Cartesian axes. The magnitude $v$ and angle $\theta$ provide all the information needed to compute $v_x$ and $v_y$ via cosine and sine. The Pythagorean invariant guarantees that no information is lost in the process. The decomposition is exact, reversible, and valid for any vector in the Euclidean plane.