Scalar projection

In mathematics, the scalar projection of a vector a {displaystyle mathbf {a} } on (or onto) a vector b {displaystyle mathbf {b} } , also known as the scalar resolute of a {displaystyle mathbf {a} } in the direction of b {displaystyle mathbf {b} } , is given by: In mathematics, the scalar projection of a vector a {displaystyle mathbf {a} } on (or onto) a vector b {displaystyle mathbf {b} } , also known as the scalar resolute of a {displaystyle mathbf {a} } in the direction of b {displaystyle mathbf {b} } , is given by: where the operator ⋅ {displaystyle cdot } denotes a dot product, b ^ {displaystyle {hat {mathbf {b} }}} is the unit vector in the direction of b {displaystyle mathbf {b} } , ‖ a ‖ {displaystyle left|mathbf {a} ight|} is the length of a {displaystyle mathbf {a} } , and θ {displaystyle heta } is the angle between a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } . The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes. The scalar projection is a scalar, equal to the length of the orthogonal projection of a {displaystyle mathbf {a} } on b {displaystyle mathbf {b} } , with a negative sign if the projection has an opposite direction with respect to b {displaystyle mathbf {b} } . Multiplying the scalar projection of a {displaystyle mathbf {a} } on b {displaystyle mathbf {b} } by b ^ {displaystyle mathbf {hat {b}} } converts it into the above-mentioned orthogonal projection, also called vector projection of a {displaystyle mathbf {a} } on b {displaystyle mathbf {b} } . If the angle θ {displaystyle heta } between a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } is known, the scalar projection of a {displaystyle mathbf {a} } on b {displaystyle mathbf {b} } can be computed using When θ {displaystyle heta } is not known, the cosine of θ {displaystyle heta } can be computed in terms of a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } , by the following property of the dot product a ⋅ b {displaystyle mathbf {a} cdot mathbf {b} } : By this property, the definition of the scalar projection s {displaystyle s,} becomes: The scalar projection has a negative sign if 90 < θ ≤ 180 {displaystyle 90< heta leq 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 a 1 {displaystyle mathbf {a} _{1}} and its length ‖ a 1 ‖ {displaystyle left|mathbf {a} _{1} ight|} :