Vector resolute

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The vector resolute (also known as the vector projection) of two vectors, \mathbf{a} in the direction of \mathbf{b} (also "\mathbf{a} on \mathbf{b}"), is given by:

(\mathbf{a}\cdot\mathbf{\hat b})\mathbf{\hat b}\text{ or }(|\mathbf{a}|\cos\theta)\mathbf{\hat b}

where θ is the angle between the vectors \mathbf{b} and \mathbf{a} and \hat{\mathbf{b}} is the unit vector in the direction of \mathbf{b}.

The vector resolute is a vector, and is the orthogonal projection of the vector \mathbf{a} onto the vector \mathbf{b}. The vector resolute is also said to be a component of vector \mathbf{a} in the direction of vector \mathbf{b}.

The other component of \mathbf{a} (perpendicular to \mathbf{b}) is given by:

\mathbf{a}\ -\ (\mathbf{a}\cdot\mathbf{\hat b})\mathbf{\hat b}.

The vector resolute is also the scalar resolute multiplied by \mathbf{\hat b} (in order to convert it into a vector, or give it direction).

Vector resolute overview

If A and B are two vectors, the projection (C) of A on B is the vector that has the same slope as B with the length:

|C| = |A| \cos \theta\,

To calculate C use the definition of the dot product: A \cdot B = |A| \, |B| \cos \theta \,

Using the above equation:

|C| = |A| \cos \theta\,

Multiply and divide by | B | at the same time:

|C| = \frac {|A| |B| \cos \theta} {|B| }\,

In the resulting fraction, the top term is the same as the dot product, hence:

|C| = \frac {A \cdot B} {|B| }\,

To find the length of | C | with an unknown θ, and unknown direction, multiply it with the unit vector B:

C = \frac {A \cdot B} {|B| } \frac {B} {|B|} = \frac {A \cdot B} {|B|^2} B,

giving the final formula:

C = \frac {A \cdot B} {|B|^2} B.

Uses

The vector projection is an important operation in the Gram-Schmidt orthonormalization of vector space bases.

See also

v • d • e
Topics related to linear algebra
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Wikipedia content modification information:

  • This page was last modified on 8 November 2008, at 01:34.

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