Vectors - Projection of a Vector on a Line

The projection of a vector $\vec{a}$ on another vector $\vec{b}$ is essentially the 'shadow' of vector $\vec{a}$ cast onto the line containing vector $\vec{b}$. Visually, if you imagine a light source positioned perpendicular to $\vec{b}$, the length of the segment on $\vec{b}$ covered by the shadow of $\vec{a}$ represents the magnitude of the projection.

The scalar projection (also called the component) of $\vec{a}$ on $\vec{b}$ is a real number given by $|\vec{a}| \cos \theta$, where $\theta$ is the angle between the two vectors. If the angle is acute, the projection is positive and points in the direction of $\vec{b}$; if obtuse, it is negative and points in the opposite direction.

The dot product is the fundamental tool for calculating projections because $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$. This relationship allows us to find the scalar component without directly calculating the angle $\theta$.

A vector projection differs from a scalar projection because it is a vector quantity. It is obtained by multiplying the scalar projection by the unit vector in the direction of $\vec{b}$. Visually, this is the actual vector arrow that lies along the line of $\vec{b}$ starting from the same origin as $\vec{a}$.

When two vectors are perpendicular (orthogonal), the angle $\theta = 90^\circ$ and $\cos 90^\circ = 0$. Consequently, the projection of one onto the other is zero. Geometrically, a vertical object cast no shadow on a horizontal line when the light source is directly above it.

The projection of a vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ on the coordinate axes ($x, y, z$) are simply its components $x$, $y$, and $z$. For example, the projection of $\vec{r}$ on the $x$-axis is the distance from the origin to the point where a perpendicular line dropped from the tip of $\vec{r}$ meets the $x$-axis.