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Scalar (Components) and Vector Projections

In a Nut Shell: The scalar projection of a vector, U, in the direction of another vector,

V, is just the component of U along V. Its symbol is, comp_V U .

comp_v U = U ∙ V / |V|

Note: The result is a scalar. U ∙ V / |V| = (U) cos θ (V) / |V| = U cos θ

Here θ is the angle between the vectors U and V .

Note: For two vectors in the x-y plane the component of U along V is U cos θ

as shown below. A similar result holds for vectors in three dimensions, x-y-z.

In a Nut Shell: The vector projection of a vector, U, in the direction of another

vector, V, is just the component of U in the direction of V times the unit vector

along V. The symbol for the vector projection of U along V is proj_V U .

proj_V U = [ ( U ∙ V ) / |V| ] e_V

Note: The result is a vector.

A unit vector in the direction of V is V / |V| = e_V

So proj_V U = [ (U cos θ) ] [ V / |V| ] = [ ( U ∙ V ) / |V| ] e_V