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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 | |
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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:
Geometrically, the dot (scalar) product of two vectors a⃗ and b⃗ represents the numerical product of the magnitude of the vector a⃗ and the projection of the vector b⃗ in the direction of a⃗.
If we appoint a basic direction (for example Ox) to the first vector a⃗, we write as b⃗x the component of the vector b⃗ in the direction of a⃗. It is obvious the component of vector b⃗ perpendicular to a⃗ is denoted as b⃗y.
Also, we can appoint a letter (for example θ) to the acute angle formed by the vector a⃗ (or its extension) and the vector b⃗. Therefore, we have for the dot (scalar) product of the two given vectors:
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
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:
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