Scalar resolute

The scalar resolute, also known as the scalar projection or scalar component, of a vector  in the direction of a vector is given by:
where is the angle

Angle

In geometry and trigonometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle...

 between the vectors and and is the unit vector

Unit vector

In mathematics, a unit vector in a normed vector space is a vector whose length is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or “hat”, like this: $\{\backslash hat\{\backslash imath\}\}$ .In Euclidean space, the dot product of two unit vectors is simply the...

 in the direction of . This is also known as " on ".

For an intuitive understanding of this formula, recall from trigonometry

Trigonometry

Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees...

 that and simply rearrange the terms by multiplying both sides by .

The scalar resolute is a scalar

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

, and is the length of the orthogonal projection of the vector onto the vector , with a minus sign if the direction is opposite.

Multiplying the scalar resolute by converts it into the vector resolute

Vector resolute

The vector resolute of two vectors, in the direction of , is given by:where is the angle between the vectors and ; the operator is the dot product; and is the unit vector in the direction of ....

, a vector.