Scalar projection

If the angle ${\displaystyle \theta }$  between ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} }$  is known, the scalar projection of ${\displaystyle \mathbf {a} }$ on${\displaystyle \mathbf {b} }$  can be computed using

${\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta .}$  (${\displaystyle s=\left\|\mathbf {a} _{1}\right\|}$  in the figure)

The formula above can be inverted to obtain the angle, θ.

When ${\displaystyle \theta }$  is not known, the cosineof${\displaystyle \theta }$  can be computed in terms of ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} ,}$  by the following property of the dot product ${\displaystyle \mathbf {a} \cdot \mathbf {b} }$ :

${\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}=\cos \theta }$ 

By this property, the definition of the scalar projection ${\displaystyle s}$  becomes:

${\displaystyle s=\left\|\mathbf {a} _{1}\right\|=\left\|\mathbf {a} \right\|\cos \theta =\left\|\mathbf {a} \right\|{\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}\,}$ 

The scalar projection has a negative sign if ${\displaystyle 90^{\circ }<\theta \leq 180^{\circ }}$ . It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. More exactly, if the vector projection is denoted ${\displaystyle \mathbf {a} _{1}}$  and its length ${\displaystyle \left\|\mathbf {a} _{1}\right\|}$ :

${\displaystyle s=\left\|\mathbf {a} _{1}\right\|}$ if${\displaystyle 0^{\circ }\leq \theta \leq 90^{\circ },}$ 

${\displaystyle s=-\left\|\mathbf {a} _{1}\right\|}$ if${\displaystyle 90^{\circ }<\theta \leq 180^{\circ }.}$ 

• Scalar product
• Cross product
• Vector projection

• Dot products - www.mit.org
• Scalar projection - Flexbooks.ck12.org
• Scalar Projection & Vector Projection - medium.com
• Lesson Explainer: Scalar Projection | Nagwa