Dot Product

The dot product (also sometimes called the scalar product) is a mathematical operation that can be performed on any two vectors with the same number of elements. The result of the dot product operation will be single scalar values that is equal to the magnitude of the first vector, times the magnitude of the second vector, times the cosine of the angle between the two vectors (at least for 2 and 3 dimensional vectors).

Dot Product Equation

If the second vector in the dot product operation is a unit vector (thus having a magnitude of 1), the dot product will represent the magnitude of the first vector in the direction of the unit vector.

Dot Product with a Unit Vector
The dot product of a vector with a unit vector will give you the magnitude of the first vector in the direction of the unit vector.

As an alternative to the above method of calculating the dot product, you can also break each vector down into components take the sum of each set of components multiplied together as shown in the equation below. Either of these two methods will yield the same result.

Dot Product Equation

Finally, as with all vector operations, we can use computer tools such as the dot command in MATLAB or the Wolfram Vector Operation Calculator tool linked to in the left panel.

Worked Problems:

Question 1:

Find the dot product of force vector A with the unit vector shown with direction B.

Problem 1 Diagram

Solution:



Question 2:

Calculate the dot product of vector A with vector B by hand.

Problem 2 Diagram

Solution:



Question 3:

Calculate the dot product of vector A with vector B using MATLAB.

Problem 3 Diagram

Solution:



Question 4:

Calculate the dot product of vector A with vector B using the Wolfram Vector Operation Calculator.

Problem 3 Diagram

Solution: