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The dot product expresses the angular relationship between two vectors. The dot product is a way of multiplying two vectors to get a quantity with only magnitude. We use the dot product to compute lengths of vectors and angles between vectors. It is also called the scalar product.
In fact, a dot product is a specific inner product. An inner product is a class of operations which satisfy certain properties. The inner product is a way to multiply vectors together and get a numerical value. The inner product works for real vector spaces as well as abstract vector spaces. The dot product only works in an Euclidean space (this is your common x-y-z space).
In the general case, the dot product is the summation of the products of the vectors' components. is the actual notation for a vector. We have two vectors: A and B. Both vectors have n components. The definition of the dot product states:
- Let .
- Let .
For example, if and then, A dot B is
Real World Example
The dot product is pretty simple, and you use it without even noticing it. It's like school shopping math. When the school year approaches, you go shopping to get school supplies. You buy pencils, folders, notebooks etc all at specific prices. To compute the total cost, you multiply the price of an item by the number of that type of item you are going to buy. Then you add it all up. You are in fact using the dot product. The prices of items are components of one vector, and the quantities you are buying of those items make up another vector. To calculate the total cost of your school supplies, you just take the dot product of these vectors.
For example, suppose you need to buy 3 pencils, 4 folders, and 5 notebooks. Pencils cost $2, folders cost $0.50, and notebooks cost $1.
The quantity vector of the items is , and the price vector of the items is .
Taking the dot product of these vectors gives us
Using the formula for dot product, we have
Simplifying, we have
Properties of Dot Products
The dot product obeys the properties listed below. This holds true for all nonzero vectors. We use another notation commonly found in textbooks. The variable is now bold instead of having a hat hover over the variable. All properties work in any dimensional space.
- (holds only for scalars such as "s")
The definition of the dot product given above was:
Although, the dot product produces a numerical value it also has a geometric interpertation. Think of multiplication, it is a process like the dot product. When you multiply two numbers you get a numerical value yet in geometry multiplication means you create a rectangle with a certain height and base. The dot product relates the length of vectors and the angle between the vectors. If we know the angle between two vectors then we can write the dot product as such:
The equation above is in fact an alternative definition of the dot product. You may wonder what exactly is and how is it different from .
If , then the length of (which is also called the norm or magnitude) denoted as is .
Geometric Properties of the Dot Product
Here are the geometric properties of the dot product.
We first put restriction on . We take
- If and are nonzero vectors then if and only if .
- when .
It makes sense for to call and perpendicular (or orthogonal). This applies to two-dimensional and three-dimensional anything higher is not a property but a definition of perpendicularity. Notice that if or is the zero vector then we can say the zero vector is perpendicular to every vector.
- The dot product can tell us other things based the measurement of the angle. If the angle is less than 90 degrees, the sign of the dot product is going to be positive. So, that means geometrically, our two vectors and are going more or less in the same direction. We can make and maximally positive by making them point in the same direction. Rotating and apart causes the dot product to decrease. The dot product reaches zero when and are perpendicular. If the angle is greater than 90 degrees the dot product will be negative and our vectors would point in opposite directions. The dot product simply measures how much the vectors are going along each other.
- and are two nonzero vectors. Imagine a perpendicular line drawn from the head of to the line through . Then this is a length called the projection of b onto a. This is written as .
- http://mathworld.wolfram.com/DotProduct.html Accessed June 13 2011.
- http://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx Accessed June 13 2011.
- Coxson, Pamela. "Introducing Linear Algebra to Middle School Students" January 16, 2010. faculty.pepperdine.edu/dstrong/LinearAlgebra/2010/JMM2010Coxson.pdf