The Full Wiki

Vector projection: Wikis

Advertisements
  

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

Encyclopedia

From Wikipedia, the free encyclopedia

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}; the operator \cdot is the dot product; 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:(By triangle addition of vectors)

\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).e.g. in this case scalar resolute is (\mathbf{a}\cdot\mathbf{\hat b}) .

Contents

Vector resolute overview

Dot Product.svg

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 following property 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 \cdot B} B,

giving the final formula:

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

Matrix representation

The orthogonal projection can be represented by a projection matrix. To project a vector onto the unit vector a = (ax, ay, az), it would need to be multiplied with this projection matrix:

 P_a = a a^T = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} = \begin{bmatrix} a_1^2 & a_1 a_2 & a_1 a_3 \ a_1 a_2 & a_2^2 & a_2 a_3 \ a_1 a_3 & a_2 a_3 & a_3^2 \ \end{bmatrix}.

Vector projection

To project a certain vector onto an other vector, the length of the projection on this new vector can be calculated by the vector product of these two vectors.

eg. When a 3 dimensional vector eg. A=[3 4 5] has to be projected onto an other 3 dimensional vector eg. B=[6 7 8], the size of the projection is equal to the scalar matrix product A * B'. = 3 * 6 + 4 * 7 + 5 * 8 = 86.This is the length of the projection on the vector B. To express this as a vector we have to multiply this result with the unity vector of Eb which is equal to B/|B|=[6 7 8]/12.2 =[0.4915 0.5735 0.6554]. So the projection vector is [0.4915 0.5735 0.6554] * 86=[42.2690 49.3210 56.3644]

Uses

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

See also

Advertisements

Advertisements






Got something to say? Make a comment.
Your name
Your email address
Message