- Vector resolute
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}cdotmathbf{hat b})mathbf{hat b} or mathbf{a}|cos heta)mathbf{hat b}
where heta is the
angle between the vectors mathbf{b} and mathbf{a} and hat{mathbf{b is theunit 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}cdotmathbf{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 heta
To calculate C use the definition of the dot product:A cdot B = |A| , |B| cos heta ,
Using the above equation:
C| = |A| cos heta
Multiply and divide by B| at the same time:
C| = frac = 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
orthonormal ization ofvector space bases.ee also
*
Scalar resolute
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