Vector resolute

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

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

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

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

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

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

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


[edit] 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 | cosθ


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

Using the above equation:

| C | = | A | cosθ


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

[edit] Uses

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

[edit] See also

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