Scalar projection

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If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection.
Vector projection of a on b (a1), and vector rejection of a from b (a2).

In mathematics, the scalar projection of a vector \mathbf{a} on (or onto) a vector \mathbf{b}, also known as the scalar resolute or scalar component of \mathbf{a} in the direction of \mathbf{b}, is given by:

s = |\mathbf{a}|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b},

where the operator \cdot denotes a dot product, \hat{\mathbf{b}} is the unit vector in the direction of \mathbf{b}, |\mathbf{a}| is the length of \mathbf{a}, and \theta is the angle between \mathbf{a} and \mathbf{b}.

The scalar projection is a scalar, equal to the length of the orthogonal projection of \mathbf{a} on \mathbf{b}, with a minus sign if the projection has an opposite direction with respect to \mathbf{b}.

Multiplying the scalar projection of \mathbf{a} on \mathbf{b} by \mathbf{\hat b} converts it into the above-mentioned orthogonal projection, also called vector projection of \mathbf{a} on \mathbf{b}.

Definition based on angle θ[edit]

If the angle \theta between \mathbf{a} and \mathbf{b} is known, the scalar projection of \mathbf{a} on \mathbf{b} can be computed using

 s = |\mathbf{a}| \cos \theta .

Definition in terms of a and b[edit]

When \theta is not known, the cosine of \theta can be computed in terms of \mathbf{a} and \mathbf{b}, by the following property of the dot product  \mathbf{a} \cdot \mathbf{b}:

 \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{a}| \, |\mathbf{b}|} = \cos \theta \,

By this property, the definition of the scalar projection s \, becomes:

s = |\mathbf{a}| \cos \theta = |\mathbf{a}| \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{a}| \, |\mathbf{b}|} = \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{b}| }\,

Properties[edit]

The scalar projection has a negative sign if 90 < \theta \le 180 degrees. It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. More exactly, if the vector projection is denoted \mathbf{a}_1 and its length |\mathbf{a}_1|:

s =  |\mathbf{a}_1| if 0  < \theta \le 90 degrees,
s = -|\mathbf{a}_1| if 90 < \theta \le 180 degrees.

See also[edit]