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In addition to the revision notes for Dot (Scalar) Product of Two Vectors on this page, you can also access the following Vectors and Scalars learning resources for Dot (Scalar) Product of Two Vectors

Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
---|---|---|---|---|---|---|

2.4 | Dot (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

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 b*⃗*_{x} 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 b*⃗*_{y}.

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*⃗*| ∙ |b*⃗*_{x}|

= |a*⃗*| ∙ |b*⃗* | ∙ cos θ

= |a

= |a

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*⃗* = x_{a} ∙ x_{b} + y_{a} ∙ y_{b}

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:

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

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- Continuing learning vectors and scalars - read our next physics tutorial: Rounding and Significant Figures

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