Vector projection

The scalar projection of a on b is a scalar equal to

$${\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta ,}$$

where θ is the angle between a and b.

A scalar projection can be used as a scale factor to compute the corresponding vector projection.

The vector projection of a on b is a vector whose magnitude is the scalar projection of a on b with the same direction as b. Namely, it is defined as

$${\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} =(\left\|\mathbf {a} \right\|\cos \theta )\mathbf {\hat {b}} }$$

where ${\displaystyle a_{1}}$ is the corresponding scalar projection, as defined above, and ${\displaystyle \mathbf {\hat {b}} }$ is the unit vector with the same direction as b:

$${\displaystyle \mathbf {\hat {b}} ={\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}}$$

By definition, the vector rejection of a on b is:

$${\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}$$

Hence,

$${\displaystyle \mathbf {a} _{2}=\mathbf {a} -\left(\left\|\mathbf {a} \right\|\cos \theta \right)\mathbf {\hat {b}} }$$

By the above-mentioned property of the dot product, the definition of the scalar projection becomes:[1]

$${\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}.}$$

In two dimensions, this becomes

$${\displaystyle a_{1}={\frac {\mathbf {a} _{x}\mathbf {b} _{x}+\mathbf {a} _{y}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}$$

Similarly, the definition of the vector projection of a onto b becomes:[1]

$${\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}},}$$

which is equivalent to either

$${\displaystyle \mathbf {a} _{1}=\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ,}$$

or[3]

$${\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}$$

In two dimensions, the scalar rejection is equivalent to the projection of a onto ${\displaystyle \mathbf {b} ^{\perp }={\begin{pmatrix}-\mathbf {b} _{y}&\mathbf {b} _{x}\end{pmatrix}}}$, which is ${\displaystyle \mathbf {b} ={\begin{pmatrix}\mathbf {b} _{x}&\mathbf {b} _{y}\end{pmatrix}}}$ rotated 90° to the left. Hence,

$${\displaystyle a_{2}=\left\|\mathbf {a} \right\|\sin \theta ={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} _{y}\mathbf {b} _{x}-\mathbf {a} _{x}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}$$

Such a dot product is called the "perp dot product."[4]

By definition,

$${\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}$$

Hence,

$${\displaystyle \mathbf {a} _{2}=\mathbf {a} -{\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }.}$$

By using the Scalar rejection using the perp dot product this gives

$${\displaystyle \mathbf {a} _{2}={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\mathbf {b} \cdot \mathbf {b} }}\mathbf {b} ^{\perp }}$$

The scalar projection a on b is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. It coincides with the length ‖c‖ of the vector projection if the angle is smaller than 90°. More exactly:

• a₁ = ‖a₁‖ if 0° ≤ θ ≤ 90°,
• a₁ = −‖a₁‖ if 90° < θ ≤ 180°.

The vector projection of a on b is a vector a₁ which is either null or parallel to b. More exactly:

• a₁ = 0 if θ = 90°,
• a₁ and b have the same direction if 0° ≤ θ < 90°,
• a₁ and b have opposite directions if 90° < θ ≤ 180°.

The vector rejection of a on b is a vector a₂ which is either null or orthogonal to b. More exactly:

• a₂ = 0 if θ = 0° or θ = 180°,
• a₂ is orthogonal to b if 0 < θ < 180°,