Scalar projection

Diagram of the scalar projection in two dimensions.

The scalar projection, also known as the scalar resolute or scalar component, of a vector $\mathbf{b}$ in the direction of a vector $\mathbf{a}$ (or scalar projection of $\mathbf{b}$ on $\mathbf{a}$) is given by:

$\mathbf{b}\cdot\mathbf{\hat a} = |\mathbf{b}|\cos\theta$

where the operator $\cdot$ denotes a dot product, $\hat{\mathbf{a}}$ is the unit vector in the direction of $\mathbf{a}$, $|\mathbf{b}|$ is the length of $\mathbf{b}$, and θ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

For an intuitive^{[citation needed]} understanding of this formula, recall from trigonometry that $\cos\theta = \frac{\mathbf{b}\cdot\mathbf{\hat a}} {|\mathbf{b}|}$ and simply rearrange the terms by multiplying both sides by $|\mathbf{b}|$.

The scalar projection is a scalar, and is the length of the orthogonal projection of the vector $\mathbf{b}$ onto the vector $\mathbf{a}$, with a minus sign if the direction is opposite.

Multiplying the scalar projection by $\mathbf{\hat a}$ converts it into the vector projection, a vector.

See also

• scalar product
• cross product