{"history":[],"article":{"id":"2b00bed2-803f-47f6-ac6e-adfff3ada23f","title":"Vector projection","slug":"vector-projection","content":":::summary\n:::\n\n# \nThe **[vector](https://bliptext.com/articles/vector-geometry) projection** (also known as the **[vector](https://bliptext.com/articles/vector-geometry) component** or **[vector](https://bliptext.com/articles/vector-geometry) resolution**) of a [vector](https://bliptext.com/articles/vector-geometry)\na\non (or onto) a nonzero vector\nb\nis the [orthogonal projection](https://bliptext.com/articles/orthogonal-projection) of\na\nonto a [straight line](https://bliptext.com/articles/straight-line) parallel to\nb\n. The projection of\na\nonto\nb\nis often written as or\na∥b\n.\nThe vector component or vector resolute of\na\n[perpendicular](https://bliptext.com/articles/perpendicular) to\nb\n, sometimes also called the **vector rejection** of\na\nfrom\nb\n(denoted or\na⊥b\n), is the orthogonal projection of\na\nonto the [plane](https://bliptext.com/articles/plane-geometry) (or, in general, hyper[plane](https://bliptext.com/articles/plane-geometry)) that is [orthogonal](https://bliptext.com/articles/orthogonal) to\nb\n. Since both and are vectors, and their sum is equal to\na\n, the rejection of\na\nfrom\nb\nis given by:\n[[File:Projection and rejection.png|thumb|200px|Projection of\na\non\nb\n(**a**1), **a**nd rejection of\na\nfrom\nb\n(**a**2).]]\n[[File:Projection and rejection 2.png|thumb|248px|When\n90° < θ ≤ 180°\n,\na1\nhas an opposite direction with respect to\nb\n.]] To simplify notation, this article defines and\nThus, the vector is parallel to \\mathbf{b}, the vector is orthogonal to \\mathbf{b}, and\nThe projection of\na\nonto\nb\ncan be decomposed into a direction and a scalar magnitude by writing it as where a_1 is a scalar, called the [scalar projection](https://bliptext.com/articles/scalar-projection) of\na\nonto\nb\n, and\nb̂\nis the [unit vector](https://bliptext.com/articles/unit-vector) in the direction of\nb\n. The scalar projection is defined as\nwhere the oper**a**tor **⋅** denotes **a** [dot product](https://bliptext.com/articles/dot-product), ‖**a**‖ is the [length](https://bliptext.com/articles/euclidean-norm) of\na\n, and θ is the [angle](https://bliptext.com/articles/angle) between\na\nand\nb\n. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is [opposite](https://bliptext.com/articles/opposite-vector) to the direction of\nb\n, that is, if the angle between the vectors is more than 90 degrees.\nThe vector projection can be calculated using the dot product of \\mathbf{a} and \\mathbf{b} as:\n\n# Notation\nThis article uses the convention that vectors are denoted in a bold font (e.g.\na1\n), and scalars are written in normal font (e.g. a1).\nThe dot product of vectors\na\nand\nb\nis written as, the norm of\na\nis written ‖**a**‖, the **a**ngle between\na\nand\nb\nis denoted θ.\n\n# Definitions based on angle θ\n\n## Scalar projection\nThe scalar projection of\na\non\nb\nis a scalar equal to\nwhere θ is the angle between\na\nand\nb\n.\nA scalar projection can be used as a [scale factor](https://bliptext.com/articles/scale-factor) to compute the corresponding vector projection.\n\n## Vector projection\nThe vector projection of\na\non\nb\nis a vector whose magnitude is the scalar projection of\na\non\nb\nwith the same direction as\nb\n. Namely, it is defined as\nwhere a_1 is the corresponding scalar projection, as defined above, and is the [unit vector](https://bliptext.com/articles/unit-vector) with the same direction as\nb\n\n## Vector rejection\nBy definition, the vector rejection of\na\non\nb\nis:\nHence,\n\n# Definitions in terms of a and b\nWhen θ is not known, the cosine of θ can be computed in terms of\na\nand\nb\n, by the following property of the [dot product](https://bliptext.com/articles/dot-product)\na ⋅ b\n\n## Scalar projection\nBy the above-mentioned property of the dot product, the definition of the scalar projection becomes:\nIn two dimensions, this becomes\n\n## Vector projection\nSimilarly, the definition of the vector projection of\na\nonto\nb\nbecomes:\nwhich is equivalent to either\nor\n\n## Scalar rejection\nIn two dimensions, the scalar rejection is equivalent to the projection of\na\nonto, which is rotated 90° to the left.\nHence,\nSuch a dot product is called the \"perp dot product.\"\n\n## Vector rejection\nBy definition,\nHence,\nBy using the Scalar rejection using the perp dot product this gives\n\n# Properties\n[[Image:Dot Product.svg|thumb|200px|right|If 0° ≤ θ ≤ 90°, as in this case, the [[scalar projection]] of\na\non\nb\ncoincides with the [length](https://bliptext.com/articles/euclidean-norm) of the vector projection.]]\n\n## Scalar projection\nThe scalar projection\na\non\nb\nis a scalar which has a negative sign if [90 degrees](https://bliptext.com/articles/right-angle) < θ ≤ [180 degrees](https://bliptext.com/articles/straight-angle).\nIt coincides with the [length](https://bliptext.com/articles/euclidean-norm)\n‖c‖\nof the vector projection if the angle is smaller than 90°.\nMore exactly:\na1 = ‖a1‖\nif\n0° ≤ θ ≤ 90°\n,\na1 = −‖a1‖\nif\n90° < θ ≤ 180°\n.\n\n## Vector projection\nThe vector projection of\na\non\nb\nis a vector\na1\nwhich is either null or parallel to\nb\n. More exactly:\na1 = 0\nif\nθ = 90°\n,\na1\nand\nb\nhave the same direction if\n0° ≤ θ < 90°\n,\na1\nand\nb\nhave opposite directions if\n90° < θ ≤ 180°\n.\n\n## Vector rejection\nThe vector rejection of\na\non\nb\nis a vector\na2\nwhich is either null or orthogonal to\nb\n. More exactly:\na2 = 0\nif\nθ = 0°\nor\nθ = 180°\n,\na2\nis orthogonal to\nb\nif\n0 < θ < 180°\n,\n\n# Matrix representation\nThe orthogonal projection can be represented by a [projection matrix](https://bliptext.com/articles/projection-linear-algebra).\nTo project a vector onto the unit vector\na = (ax, ay, az)\n, it would need to be multiplied with this projection matrix:\n\n# Uses\nThe vector projection is an important operation in the [Gram–Schmidt](https://bliptext.com/articles/gram-schmidt-process) [orthonormalization](https://bliptext.com/articles/orthonormality) of [vector space](https://bliptext.com/articles/vector-space) [bases](https://bliptext.com/articles/basis-linear-algebra).\nIt is also used in the [separating axis theorem](https://bliptext.com/articles/separating-axis-theorem) to detect whether two convex shapes intersect.\n\n# Generalizations\nSince the notions of vector [length](https://bliptext.com/articles/length) and [angle](https://bliptext.com/articles/angle) between vectors can be generalized to any n-dimensional [inner product space](https://bliptext.com/articles/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.\nIn some cases, the inner product coincides with the dot product.\nWhenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection.\nFor a three-dimensional [inner product space](https://bliptext.com/articles/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](https://bliptext.com/articles/plane-geometry), and rejection of a vector from a [plane](https://bliptext.com/articles/plane-geometry).\nThe projection of a vector on a plane is its [orthogonal projection](https://bliptext.com/articles/orthogonal-projection) on that plane.\nThe rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane.\nBoth are vectors.\nThe first is parallel to the plane, the second is orthogonal.\nFor a given vector and plane, the sum of projection and rejection is equal to the original vector.\nSimilarly, 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](https://bliptext.com/articles/[hyperplane](https://bliptext.com/articles/hyperplane)), and rejection from a [hyperplane](https://bliptext.com/articles/[hyperplane](https://bliptext.com/articles/hyperplane)).\nIn [geometric algebra](https://bliptext.com/articles/geometric-algebra), they can be further generalized to the notions of [projection and rejection](https://bliptext.com/articles/geometric-algebra) of a general multivector onto/from any invertible k-blade.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:25:08.430Z","updated_at":"2025-01-25T21:25:08.430Z","search_vector":"'project':2 'vector':1","search_content":" \n \n\n# \nThe **[vector](https://bliptext.com/articles/vector-geometry) projection** (also known as the **[vector](https://bliptext.com/articles/vector-geometry) component** or **[vector](https://bliptext.com/articles/vector-geometry) resolution**) of a [vector](https://bliptext.com/articles/vector-geometry)\na\non (or onto) a nonzero vector\nb\nis the [orthogonal projection](https://bliptext.com/articles/orthogonal-projection) of\na\nonto a [straight line](https://bliptext.com/articles/straight-line) parallel to\nb\n. The projection of\na\nonto\nb\nis often written as or\na∥b\n.\nThe vector component or vector resolute of\na\n[perpendicular](https://bliptext.com/articles/perpendicular) to\nb\n, sometimes also called the **vector rejection** of\na\nfrom\nb\n(denoted or\na⊥b\n), is the orthogonal projection of\na\nonto the [plane](https://bliptext.com/articles/plane-geometry) (or, in general, hyper[plane](https://bliptext.com/articles/plane-geometry)) that is [orthogonal](https://bliptext.com/articles/orthogonal) to\nb\n. Since both and are vectors, and their sum is equal to\na\n, the rejection of\na\nfrom\nb\nis given by:\n[[File:Projection and rejection.png|thumb|200px|Projection of\na\non\nb\n(**a**1), **a**nd rejection of\na\nfrom\nb\n(**a**2).]]\n[[File:Projection and rejection 2.png|thumb|248px|When\n90° < θ ≤ 180°\n,\na1\nhas an opposite direction with respect to\nb\n.]] To simplify notation, this article defines and\nThus, the vector is parallel to \\mathbf{b}, the vector is orthogonal to \\mathbf{b}, and\nThe projection of\na\nonto\nb\ncan be decomposed into a direction and a scalar magnitude by writing it as where a_1 is a scalar, called the [scalar projection](https://bliptext.com/articles/scalar-projection) of\na\nonto\nb\n, and\nb̂\nis the [unit vector](https://bliptext.com/articles/unit-vector) in the direction of\nb\n. The scalar projection is defined as\nwhere the oper**a**tor **⋅** denotes **a** [dot product](https://bliptext.com/articles/dot-product), ‖**a**‖ is the [length](https://bliptext.com/articles/euclidean-norm) of\na\n, and θ is the [angle](https://bliptext.com/articles/angle) between\na\nand\nb\n. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is [opposite](https://bliptext.com/articles/opposite-vector) to the direction of\nb\n, that is, if the angle between the vectors is more than 90 degrees.\nThe vector projection can be calculated using the dot product of \\mathbf{a} and \\mathbf{b} as:\n\n# Notation\nThis article uses the convention that vectors are denoted in a bold font (e.g.\na1\n), and scalars are written in normal font (e.g. a1).\nThe dot product of vectors\na\nand\nb\nis written as, the norm of\na\nis written ‖**a**‖, the **a**ngle between\na\nand\nb\nis denoted θ.\n\n# Definitions based on angle θ\n\n## Scalar projection\nThe scalar projection of\na\non\nb\nis a scalar equal to\nwhere θ is the angle between\na\nand\nb\n.\nA scalar projection can be used as a [scale factor](https://bliptext.com/articles/scale-factor) to compute the corresponding vector projection.\n\n## Vector projection\nThe vector projection of\na\non\nb\nis a vector whose magnitude is the scalar projection of\na\non\nb\nwith the same direction as\nb\n. Namely, it is defined as\nwhere a_1 is the corresponding scalar projection, as defined above, and is the [unit vector](https://bliptext.com/articles/unit-vector) with the same direction as\nb\n\n## Vector rejection\nBy definition, the vector rejection of\na\non\nb\nis:\nHence,\n\n# Definitions in terms of a and b\nWhen θ is not known, the cosine of θ can be computed in terms of\na\nand\nb\n, by the following property of the [dot product](https://bliptext.com/articles/dot-product)\na ⋅ b\n\n## Scalar projection\nBy the above-mentioned property of the dot product, the definition of the scalar projection becomes:\nIn two dimensions, this becomes\n\n## Vector projection\nSimilarly, the definition of the vector projection of\na\nonto\nb\nbecomes:\nwhich is equivalent to either\nor\n\n## Scalar rejection\nIn two dimensions, the scalar rejection is equivalent to the projection of\na\nonto, which is rotated 90° to the left.\nHence,\nSuch a dot product is called the \"perp dot product.\"\n\n## Vector rejection\nBy definition,\nHence,\nBy using the Scalar rejection using the perp dot product this gives\n\n# Properties\n[[Image:Dot Product.svg|thumb|200px|right|If 0° ≤ θ ≤ 90°, as in this case, the [[scalar projection]] of\na\non\nb\ncoincides with the [length](https://bliptext.com/articles/euclidean-norm) of the vector projection.]]\n\n## Scalar projection\nThe scalar projection\na\non\nb\nis a scalar which has a negative sign if [90 degrees](https://bliptext.com/articles/right-angle) < θ ≤ [180 degrees](https://bliptext.com/articles/straight-angle).\nIt coincides with the [length](https://bliptext.com/articles/euclidean-norm)\n‖c‖\nof the vector projection if the angle is smaller than 90°.\nMore exactly:\na1 = ‖a1‖\nif\n0° ≤ θ ≤ 90°\n,\na1 = −‖a1‖\nif\n90° < θ ≤ 180°\n.\n\n## Vector projection\nThe vector projection of\na\non\nb\nis a vector\na1\nwhich is either null or parallel to\nb\n. More exactly:\na1 = 0\nif\nθ = 90°\n,\na1\nand\nb\nhave the same direction if\n0° ≤ θ < 90°\n,\na1\nand\nb\nhave opposite directions if\n90° < θ ≤ 180°\n.\n\n## Vector rejection\nThe vector rejection of\na\non\nb\nis a vector\na2\nwhich is either null or orthogonal to\nb\n. More exactly:\na2 = 0\nif\nθ = 0°\nor\nθ = 180°\n,\na2\nis orthogonal to\nb\nif\n0 < θ < 180°\n,\n\n# Matrix representation\nThe orthogonal projection can be represented by a [projection matrix](https://bliptext.com/articles/projection-linear-algebra).\nTo project a vector onto the unit vector\na = (ax, ay, az)\n, it would need to be multiplied with this projection matrix:\n\n# Uses\nThe vector projection is an important operation in the [Gram–Schmidt](https://bliptext.com/articles/gram-schmidt-process) [orthonormalization](https://bliptext.com/articles/orthonormality) of [vector space](https://bliptext.com/articles/vector-space) [bases](https://bliptext.com/articles/basis-linear-algebra).\nIt is also used in the [separating axis theorem](https://bliptext.com/articles/separating-axis-theorem) to detect whether two convex shapes intersect.\n\n# Generalizations\nSince the notions of vector [length](https://bliptext.com/articles/length) and [angle](https://bliptext.com/articles/angle) between vectors can be generalized to any n-dimensional [inner product space](https://bliptext.com/articles/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.\nIn some cases, the inner product coincides with the dot product.\nWhenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection.\nFor a three-dimensional [inner product space](https://bliptext.com/articles/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](https://bliptext.com/articles/plane-geometry), and rejection of a vector from a [plane](https://bliptext.com/articles/plane-geometry).\nThe projection of a vector on a plane is its [orthogonal projection](https://bliptext.com/articles/orthogonal-projection) on that plane.\nThe rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane.\nBoth are vectors.\nThe first is parallel to the plane, the second is orthogonal.\nFor a given vector and plane, the sum of projection and rejection is equal to the original vector.\nSimilarly, 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](https://bliptext.com/articles/hyperplane)](https://bliptext.com/articles/[hyperplane](https://bliptext.com/articles/hyperplane)), and rejection from a [[hyperplane](https://bliptext.com/articles/hyperplane)](https://bliptext.com/articles/[hyperplane](https://bliptext.com/articles/hyperplane)).\nIn [geometric algebra](https://bliptext.com/articles/geometric-algebra), they can be further generalized to the notions of [projection and rejection](https://bliptext.com/articles/geometric-algebra) of a general multivector onto/from any invertible k-blade.","revisionCount":0},"hasMore":false,"nextCursor":null}

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