The vector resolute (also known as the vector projection) of two vectors, in the direction of (also " on "), is given by:
where θ is the angle between the vectors and ; the operator is the dot product; and is the unit vector in the direction of .
The vector resolute is a vector, and is the orthogonal projection of the vector onto the vector . The vector resolute is also said to be a component of vector in the direction of vector .
The other component of (perpendicular to ) is given by:(By triangle addition of vectors)
The vector resolute is also the scalar resolute multiplied by (in order to convert it into a vector, or give it direction).e.g. in this case scalar resolute is .
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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:
To calculate C use the following property of the dot product:
Using the above equation:
Multiply and divide by  B  at the same time:
In the resulting fraction, the top term is the same as the dot product, hence:
To find the length of  C  with an unknown θ, and unknown direction, multiply it with the unit vector B:
giving the final formula:
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:
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]
The vector projection is an important operation in the GramSchmidt orthonormalization of vector space bases.

