Scalar resolute

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Diagram of the scalar projection in two dimensions.

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

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

where $θ$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ and $\hat{\mathbf{a}}$ is the unit vector in the direction of $\mathbf{a}$ . This is also known as " $\mathbf{b}$ on $\mathbf{a}$ ".

For an intuitive 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 resolute 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 resolute by $\mathbf{\hat a}$ converts it into the vector resolute, a vector.