The vector projection of a vector a on (or onto) a nonzero vector b (also known as the vector component or vector resolute of a in the direction of b) is the orthogonal projection of a onto a straight line parallel to b. It is a vector parallel to b, defined as
- $\backslash mathbf\{a\}\_1\; =\; a\_1\backslash mathbf\{\backslash hat\; b\}$
where a_{1} is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b.
In turn, the scalar projection is defined as
- $a\_1\; =\; |\backslash mathbf\{a\}|\backslash cos\backslash theta\; =\; \backslash mathbf\{a\}\backslash cdot\backslash mathbf\{\backslash hat\; b\},$
where the operator · denotes a dot product, |a| is the length of a, and θ is the angle between a and b. The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b.
The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b,^{[1]}
is the orthogonal projection of a onto the plane (or, in general, hyperplane) orthogonal to b. Both the projection a_{1} and rejection a_{2} of a vector a are vectors, and their sum is equal to a, which implies that the rejection is given by
- $\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash mathbf\{a\}\_1.$
Notation
Typically, a vector projection is denoted with an uppercase letter (e.g. a_{1}), and the corresponding scalar projection with a lowercase letter (e.g. a_{1}). In some cases, especially in handwriting, the vector projection is also denoted using a diacritic above or below the letter (e.g., $\backslash vec\{a\}\_1$ or a_{1}; see Euclidean vector representations for more details).
The vector projection of a on b and the corresponding rejection are sometimes denoted by a_{∥b} and a_{⊥b}, respectively.
Definitions based on angle θ
Scalar projection
The scalar projection of a on b is a scalar equal to
- $a\_1\; =\; |\backslash mathbf\{a\}|\; \backslash cos\; \backslash theta$
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.
Vector projection
The vector projection of a on b is a vector whose magnitude is the scalar projection of a on b and whose angle against b is either 0 or 180 degrees.
Namely, it is defined as
- $\backslash mathbf\{a\}\_1\; =\; a\_1\; \backslash mathbf\{\backslash hat\; b\}\; =\; (|\backslash mathbf\{a\}|\; \backslash cos\; \backslash theta)\; \backslash mathbf\{\backslash hat\; b\}$
where a_{1} is the corresponding scalar projection, as defined above, and b̂ is the unit vector with the same direction as b:
- $\backslash mathbf\{\backslash hat\; b\}\; =\; \backslash frac\; \{\backslash mathbf\{b\}\}\; \{|\backslash mathbf\{b\}|\}\backslash ,$
Vector rejection
By definition, the vector rejection of a on b is
- $\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash mathbf\{a\}\_1$
Hence,
- $\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; (|\backslash mathbf\{a\}|\; \backslash cos\; \backslash theta)\; \backslash mathbf\{\backslash hat\; b\}.$
Definitions in terms of a and b
When θ is not known, the cosine of θ can be computed in terms of a and b, by the following property of the dot product a · b:
- $\backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{|\backslash mathbf\{a\}|\; \backslash ,\; |\backslash mathbf\{b\}|\}\; =\; \backslash cos\; \backslash theta\; \backslash ,$
Scalar projection
By the above mentioned property of the dot product, the definition of the scalar projection becomes
- $a\_1\; =\; |\backslash mathbf\{a\}|\; \backslash cos\; \backslash theta\; =\; |\backslash mathbf\{a\}|\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{|\backslash mathbf\{a\}|\; \backslash ,\; |\backslash mathbf\{b\}|\}\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{|\backslash mathbf\{b\}|\; \}\backslash ,$
Vector projection
Similarly, the definition of the vector projection becomes
- $\backslash mathbf\{a\}\_1\; =\; a\_1\; \backslash mathbf\{\backslash hat\; b\}\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{|\backslash mathbf\{b\}|\; \}\; \backslash frac\; \{\backslash mathbf\{b\}\}\; \{|\backslash mathbf\{b\}|\},$
which is equivalent to either
- $\backslash mathbf\{a\}\_1\; =\; (\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{\backslash hat\; b\})\; \backslash mathbf\{\backslash hat\; b\},$
or^{[2]}
- $\backslash mathbf\{a\}\_1\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{|\backslash mathbf\{b\}|^2\}\{\backslash mathbf\{b\}\}\; =\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{\backslash mathbf\{b\}\; \backslash cdot\; \backslash mathbf\{b\}\}\{\backslash mathbf\{b\}\}.$
The latter formula is computationally more efficient than the former. Both require two dot products and eventually the multiplication of a scalar by a vector, but the former additionally requires a square root and the division of a vector by a scalar,^{[3]}
while the latter additionally requires only the division of a scalar by a scalar.
Vector rejection
By definition,
- $\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash mathbf\{a\}\_1$
Hence,
- $\backslash mathbf\{a\}\_2\; =\; \backslash mathbf\{a\}\; -\; \backslash frac\; \{\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\}\; \{\backslash mathbf\{b\}\; \backslash cdot\; \backslash mathbf\{b\}\}\{\backslash mathbf\{b\}\}.$
Properties
Scalar projection
The scalar projection a on b is a scalar which has a negative sign if 90 < θ ≤ 180 degrees. It coincides with the length |c| of the vector projection if the angle is smaller than 90°. More exactly:
- a_{1} = |a_{1}| if 0 ≤ θ ≤ 90 degrees,
- a_{1} = −|a_{1}| if 90 < θ ≤ 180 degrees.
Vector projection
The vector projection of a on b is a vector a_{1} which is either null or parallel to b. More exactly,
- a_{1} = 0 if θ = 90°,
- a_{1} and b have the same direction if 0 ≤ θ < 90 degrees,
- a_{1} and b have opposite directions if 90 < θ ≤ 180 degrees.
Vector rejection
The vector rejection of a on b is a vector a_{2} which is either null or orthogonal to b. More exactly,
- a_{2} = 0 if θ = 0 degrees or θ = 180 degrees,
- a_{2} is orthogonal to b if 0 < θ < 180 degrees,
Matrix representation
The orthogonal projection can be represented by a projection matrix. To project a vector onto the unit vector a = (a_{x}, a_{y}, a_{z}), it would need to be multiplied with this projection matrix:
- $P\_a\; =\; a\; a^\backslash mathrm\{T\}\; =$
\begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}
\begin{bmatrix} a_x & a_y & a_z \end{bmatrix} =
\begin{bmatrix}
a_x^2 & a_x a_y & a_x a_z \\
a_x a_y & a_y^2 & a_y a_z \\
a_x a_z & a_y a_z & a_z^2 \\
\end{bmatrix}
Uses
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.
Generalizations
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.^{[4]}
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.
See also
References
External links
- Projection of a vector onto a plane
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