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Vector Projection

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Suppose we have a vector in 2-dimensional space, and we would like to find the component of vector  in the direction of horizontal and vertical axis. Using trigonometry we can find the magnitude of vector component to be  and. The direction of these vector components are the standard unit vector of horizontal and vertical axes  and.

We can find vector projection of a vector onto other vector based on the inner product of the two vectors. Scalar projection of vector onto vector  is the magnitude of projection given as. Vector projection of vector onto vector  is the magnitude of projection times the unit vector of becomes. Geometrically, vector projection is shown in the figure below.

Example
Projection of vector onto horizontal axis is equivalent to projection of vector onto unit standard vector .  The scalar projection is. The projection vector becomes. Cosine angle between two vectors is . Thus, .

The interactive program below will give you the projection vectors and the scalar projection. You need to provide 2 vectors as the input. Random Example button will generate random input vectors.

vector x vector y