Scalar projection

Definition based on angle ''θ''

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 = \left|\mathbf{a}\right| \cos \theta . (s = \left|\mathbf{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 \mathbf{a} and \mathbf{b}, by the following property of the dot product \mathbf{a} \cdot \mathbf{b}: : \frac {\mathbf{a} \cdot \mathbf{b}} {\left|\mathbf{a}\right| \left|\mathbf{b}\right|} = \cos \theta

By this property, the definition of the scalar projection s becomes: : s = \left|\mathbf{a}_1\right| = \left|\mathbf{a}\right| \cos \theta = \left|\mathbf{a}\right| \frac {\mathbf{a} \cdot \mathbf{b}} {\left|\mathbf{a}\right| \left|\mathbf{b}\right|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left|\mathbf{b}\right| },

Properties

The scalar projection has a negative sign if 90^\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 \mathbf{a}_1 and its length \left|\mathbf{a}_1\right|:

: s = \left|\mathbf{a}_1\right| if 0^\circ \le \theta \le 90^\circ, : s = -\left|\mathbf{a}_1\right| if 90^\circ

Sources

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

References

References

1. Strang, Gilbert. (2016). "Introduction to linear algebra". Cambridge press.