Scalar projection explained

a on (or onto) a vector b , also known as the scalar resolute of a in the direction of b , is given by:s = \left\| a \right\|\cos\theta = a ⋅ \hatb ,

where the operator

⋅ denotes a dot product, \hat{ b } is the unit vector in the direction of b , \left\| a \right\| is the length of a , and \theta is the angle between a and b .^[1]

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 on b , with a negative sign if the projection has an opposite direction with respect to b .

Multiplying the scalar projection of

a on b by \hatb converts it into the above-mentioned orthogonal projection, also called vector projection of a on b .

Definition based on angle θ

\theta between a and b is known, the scalar projection of a on b can be computed usings = \left\| a \right\| \cos \theta . (s = \left\| a _{1\right\|} in the figure)

The formula above can be inverted to obtain the angle, θ.

Definition in terms of a and b

When

\theta is not known, the cosine of \theta can be computed in terms of a and b , by the following property of the dot product a ⋅ b :(a ⋅ b)/(\left\| a \right\| \left\| b \right\|) = \cos \theta

By this property, the definition of the scalar projection

s becomes:s = \left\| a _{1\right\|} = \left\| a \right\| \cos \theta = \left\| a \right\| (a ⋅ b)/(\left\| a \right\| \left\| b \right\|) = (a ⋅ b)/(\left\| b \right\|)

Properties

The scalar projection has a negative sign if

90 ^{\circ} < \theta \le 180 ^{\circ} . 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} and its length \left\| a _{1\right\|} :s = \left\| a _{1\right\|} if 0 ^{\circ} \le \theta \le 90 ^{\circ,} s = - \left\| a _{1\right\|} if 90 ^{\circ} < \theta \le 180 ^{\circ.}

See also

• Scalar product
• Cross product
• Vector projection

Sources

• Dot products - www.mit.org
• Scalar projection - Flexbooks.ck12.org
• Scalar Projection & Vector Projection - medium.com
• Lesson Explainer: Scalar Projection | Nagwa

Notes and References

1. Book: Strang, Gilbert . Introduction to linear algebra . 2016 . Cambridge press . 978-0-9802327-7-6 . 5th . Wellesley.