Dot (Scalar) Product of Two Vectors

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2.4Dot (Scalar) Product of Two Vectors


In these revision notes for Dot (Scalar) Product of Two Vectors, we cover the following key points:

  • The meaning of "dot (scalar) product" of two vectors (both geometrically and conceptually)
  • How to calculate the dot (scalar) product of two vectors? (two methods)
  • Some applications of dot (scalar) product in Physics

Dot (Scalar) Product of Two Vectors Revision Notes

Geometrically, the dot (scalar) product of two vectors a and b represents the numerical product of the magnitude of the vector a and the projection of the vector b in the direction of a.

If we appoint a basic direction (for example Ox) to the first vector a, we write as bx the component of the vector b in the direction of a. It is obvious the component of vector b perpendicular to a is denoted as by.

Also, we can appoint a letter (for example θ) to the acute angle formed by the vector a (or its extension) and the vector b. Therefore, we have for the dot (scalar) product of the two given vectors:

c = a ∙ b
= |a| ∙ |bx|
= |a| ∙ |b | ∙ cos θ

The result c is a scalar because we found it by multiplying two vector magnitudes (which are simply numbers) and the cosine of an angle (which is a number as well).

If the coordinates of the two vectors are known, it is much easier to calculate their dot product by using the formula

a ∙ b = xa ∙ xb + ya ∙ yb

This formula is particularly useful when none of vectors lies according to a main direction (axis).

There are many applications of dot (scalar) product of two vectors in Physics. Some of them include:

  1. Work as a dot product of Force and Displacement.
  2. Power as a dot product of Force and Velocity.
  3. Calculation of the angle between two forces acting on the same object.
  4. Kinetic Energy as a dot product of linear momentum and velocity.

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