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Introducing vectors for engineering applications
Introducing vectors for engineering applications

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4 Scalar product of vectors

This section explores a way to multiply two vectors, which is called the scalar product or the dot product. It is called the scalar product, because when using this method of multiplication, the result is a scalar quantity. It is also called the dot product because it is written using the symbol ‘dot operator’, for example, the dot product of vectors bold a and bold b is written a dot operator b.

Let’s start by considering what multiplication might mean in the context of vectors. For scalar quantities, multiplication can be thought of as repeated counting. For example, four multiplication three can mean sum with 3 summands four plus four plus four. Alternatively, we can think of multiplication as taking a magnitude and growing it. For example, four multiplication three can mean taking a magnitude of 4 and making it 3 times larger. For the scalar product of vectors, it is useful to think of multiplication in terms of growth.

Vectors have direction as well as magnitude, and if we consider vector operations in terms of growth, then we can describe them as follows.

  • Adding vectors: accumulate growth from several vectors.

  • Scalar multiplication: make an existing vector grow.

  • Scalar product: apply the directed growth of one vector to another vector. The result is how much stronger we have made the original.

For example, if we are talking about force vectors, then the scalar product gives us a measure of how much push one vector can give to another. This cannot be just a matter of multiplying the magnitude of the vectors, because their directions need to be taken into consideration. So the scalar product is a multiplication operation that takes into consideration the directions of vectors. With this concept in mind, let’s look at some examples.