Vector projection#Vector projection

This article uses the convention that vectors are denoted in a bold font (e.g. {{math|a₁}}), and scalars are written in normal font (e.g. a₁).

The dot product of vectors {{math|a}} and {{math|b}} is written as $\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{b\}$, the norm of {{math|a}} is written ‖a‖, the angle between {{math|a}} and {{math|b}} is denoted θ.

{{Main|Scalar projection}}

The scalar projection of {{math|a}} on {{math|b}} is a scalar equal to

$$a\_1\; =\; \backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\; \backslash cos\; \backslash theta\; ,$$

where θ is the angle between {{math|a}} and {{math|b}}.

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

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

$$\backslash mathbf\{a\}\_1\; =\; a\_1\; \backslash mathbf\{\backslash hat\; b\}\; =\; (\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\; \backslash cos\; \backslash theta)\; \backslash mathbf\{\backslash hat\; b\}$$

where $a\_1$ is the corresponding scalar projection, as defined above, and $\backslash mathbf\{\backslash hat\; b\}$ is the unit vector with the same direction as {{math|b}}:

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

By definition, the vector rejection of {{math|a}} on {{math|b}} is:

$$\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash mathbf\{a\}\_1$$

Hence,

$$\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash left(\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\; \backslash cos\; \backslash theta\backslash right)\; \backslash mathbf\{\backslash hat\; b\}$$

When {{mvar|θ}} is not known, the cosine of {{mvar|θ}} can be computed in terms of {{math|a}} and {{math|b}}, by the following property of the dot product {{math|a ⋅ b}}

$$\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\; =\; \backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\; \backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\; \backslash cos\; \backslash theta$$

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

In two dimensions, this becomes

$$a\_1\; =\; \backslash frac\; \{\backslash mathbf\{a\}\_x\; \backslash mathbf\{b\}\_x\; +\; \backslash mathbf\{a\}\_y\; \backslash mathbf\{b\}\_y\}\; \{\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\}.$$

Similarly, the definition of the vector projection of {{math|a}} onto {{math|b}} becomes:

$$\backslash mathbf\{a\}\_1\; =\; a\_1\; \backslash mathbf\{\backslash hat\; b\}\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\; \}\; \backslash frac\; \{\backslash mathbf\{b\}\}\; \{\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\},$$

which is equivalent to either

$$\backslash mathbf\{a\}\_1\; =\; \backslash left(\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{\backslash hat\; b\}\backslash right)\; \backslash mathbf\{\backslash hat\; b\},$$

or{{cite web|url=http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html | title=Dot Products and Projections}} {{Dead link|date=January 2025}}

$$\backslash mathbf\{a\}\_1\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |^2\}\{\backslash mathbf\{b\}\}\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{\backslash mathbf\{b\}\; \backslash cdot\; \backslash mathbf\{b\}\}\{\backslash mathbf\{b\}\}\; ~\; .$$

In two dimensions, the scalar rejection is equivalent to the projection of {{math|a}} onto , which is rotated 90° to the left. Hence,

$$a\_2\; =\; \backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\; \backslash sin\; \backslash theta\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}^\backslash perp\}\; \{\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\}\; =\; \backslash frac\; \{\backslash mathbf\{a\}\_y\; \backslash mathbf\{b\}\_x\; -\; \backslash mathbf\{a\}\_x\; \backslash mathbf\{b\}\_y\}\; \{\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\; \}.$$

Such a dot product is called the "perp dot product."

By definition,

$$\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash mathbf\{a\}\_1$$

Hence,

$$\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{\backslash mathbf\{b\}\; \backslash cdot\; \backslash mathbf\{b\}\}\{\backslash mathbf\{b\}\}.$$

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

$$\backslash mathbf\{a\}\_2\; =\; \backslash frac\{\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{b\}^\backslash perp\}\{\backslash mathbf\{b\}\backslash cdot\backslash mathbf\{b\}\}\backslash mathbf\{b\}^\backslash perp$$

Image:Dot Product.svg of {{math|a}} on {{math|b}} coincides with the length of the vector projection.|center]]

{{Main|Scalar projection}}

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

• {{math|1=a₁ = ‖a₁‖}} if {{math|0° ≤ θ ≤ 90°}},
• {{math|1=a₁ = −‖a₁‖}} if {{math|90° < θ ≤ 180°}}.

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

• {{math|1=a₁ = 0}} if {{math|1=θ = 90°}},
• {{math|a₁}} and {{math|b}} have the same direction if {{math|0° ≤ θ < 90°}},
• {{math|a₁}} and {{math|b}} have opposite directions if {{math|90° < θ ≤ 180°}}.

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

• {{math|1=a₂ = 0}} if {{math|1=θ = 0°}} or {{math|1=θ = 180°}},
• {{math|1=a₂}} is orthogonal to {{math|b}} if {{math|1=0 < θ < 180°}},

The orthogonal projection can be represented by a projection matrix. To project a vector onto the unit vector {{math|1=a = (a_x, a_y, a_z)}}, it would need to be multiplied with this projection matrix:

The vector projection is an important operation in the Gram–Schmidt orthonormalization of vector space bases. It is also used in the separating axis theorem to detect whether two convex shapes intersect.

Since the notions of vector length and angle between vectors can be generalized to any n-dimensional inner product space, this is also true for the notions of orthogonal projection of a vector, projection of a vector onto another, and rejection of a vector from another.

In some cases, the inner product coincides with the dot product. Whenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. For a three-dimensional inner product space, the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a plane, and rejection of a vector from a plane. M.J. Baker, 2012. [http://www.euclideanspace.com/maths/geometry/elements/plane/lineOnPlane/index.htm Projection of a vector onto a plane.] Published on www.euclideanspace.com. The projection of a vector on a plane is its orthogonal projection on that plane. The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal.

For a given vector and plane, the sum of projection and rejection is equal to the original vector. Similarly, for inner product spaces with more than three dimensions, the notions of projection onto a vector and rejection from a vector can be generalized to the notions of projection onto a hyperplane, and rejection from a hyperplane. In geometric algebra, they can be further generalized to the notions of projection and rejection of a general multivector onto/from any invertible k-blade.

• Scalar projection
• Vector notation

{{reflist}}

• [http://www.euclideanspace.com/maths/geometry/elements/plane/lineOnPlane/index.htm Projection of a vector onto a plane]

{{Linear algebra}}

projection

Category:Transformation (function)

Category:Functions and mappings