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Vector multiplication – Types, Process, and Examples
Vector multiplication helps us understand how two vectors behave when combined. This vector operation has an extensive application in physics, engineering, and astronomy, so we need to learn about these techniques, especially if we study higher maths.
Vector multiplication covers two important techniques in vector operations: scalar product and cross product.
Learning about vector multiplication can also help us refresh our knowledge of vectors and vector application topics.
This article will discuss the two types of vector multiplication and learn the difference between the two. Make sure to keep your notes on the following concepts handy since we may have to brush up on them while learning about vector multiplication.
- Understand the different components that make up a vector.
- Review how we add and subtract vectors.
- Understand how a scalar factor affects a given vector.
For now, let’s go ahead and learn the two important techniques of vector multiplication.
How to multiply vectors?
When we multiply two or more vectors, it is important to determine whether we want a product that has a scalar quantity or vector quantity. The technique we’ll need to apply depends on our answer to that question.
There are actually three possible products in vector multiplication: vector multiplied by a scalar factor, the dot (or scalar) product, and the cross (or vector) product.
- At this point, we should have learned about distributing scalar factors to a vector, and that’s the first procedure. Make sure the check the links we’ve included in the first section.
- The dot product or known as the scalar product, as you have guessed, returns a scalar quantity.
- Similarly, the cross product returns a vector quantity.
Our discussion will focus on the latter two techniques: dot and cross products. These names will help you identify the operation that needs to be done since their operators represent a dot (
What are the vector multiplication rules?
The two products will have different results and processes. This is why we need to understand what the dot and cross products represent.
Dot Product and Its Rules
The dot product represents the projection of one vector onto another vector. Let’s say we have
Here’s a visualization of what happens when we find the dot product of
This means that the dot product of
The resulting product is scalar, so the dot product is also known as the scalar product. Here are some important properties to keep in mind about the scalar product:
- The scalar product is commutative:
. - The scalar product is distributive:
. - The scalar product of two perpendicular vectors will always be equal to
(that’s because is equal to ).
There’s another way to calculate the two vectors’ dot product: we multiply their respective Cartesian components. Let’s say
We can calculate these two ways for the dot product (or scalar product) of two vectors. Now, let’s move on to the second important result of vector multiplication: the cross or vector product.
Cross Product and Its Rules
You might already see a pattern here- for cross products, we use the operator
A great way to visualize two vectors’ cross product is by determining the area of the parallelogram by the vectors.
From this, we can see that the cross product of
Keep in mind that
Here are some important properties of vector or cross products that may come in handy:
- Vector or cross products are anti-commutative:
. - Vector product is distributive over addition:
. - The cross product of two parallel vectors will always be equal to
.
As with dot products, we can also find the vector product of two vectors given their Cartesian forms. Using the same unit vectors, we have
An important property to observe is the vector products of the three-unit vectors. We’ve summarized the products for you here:
We can develop the cross-product formula by distributing the two vectors over addition and using the cross-products of the unit vectors. However, to keep this discussion straightforward, here’s how we can calculate for the vector product of
If you have already studied determinants in the past, there’s another way of understanding the vector product, and you may help you remember the operation as well.
- We can write the unit vectors on the first array.
- The two vectors’ components in the next two arrays.
- To find the coefficient before
, we find the determinant of the remaining matrix once we cover the row and column that contains . - Apply a similar process to find the components for
and .
Regardless of the approach, it should still return the same results. That’s all we need to learn so far about vector multiplication, so why don’t we go ahead and try out these problems below to better understand cross-products and dot products?
Example 1
In Physics, the work done on an object can be calculated using the formula,
What is the work done on the object with the following components?
a.
b.
c.
Solution
This problem requires us to multiply the magnitudes of
We’ll use the same formula for all three values so, we’ll go ahead and summarize the calculations in one table.
a. | ||
b. | ||
c. |
This is a great way to apply our dot product formula and also get a glimpse of one of the many applications of vector multiplication.
Example 2
Find the expressions for
Solution
When given the Cartesian forms of the vectors, we can still find their dot and cross products.
- We can multiply the corresponding components’ coefficients then add the products to find the dot product.
- For the cross product with the form,
, we distribute the terms algebraically and cancel out cross products of the same unit vectors.
Why don’t we work with
The dot product will be straightforward – we multiply the coefficients before
This means that the dot product of
Since the vector product is distributive over addition, what we can do is distribute the terms algebraically and cancel out
Recall that identical vectors’ cross product will be equal to zero, so let’s use this to cancel out terms. Also, review the guide we provided for the cross-products of other unit vectors.
Hence, we’ve shown how we can find the cross and dot products of two vectors of the form,
Example 3
Evaluate the following expressions given the Cartesian forms of the three vectors shown below.
a.
b.
c.
Solution
There are two ways for us to find the cross product of
Hence, we have the cross product of the two vectors:
We’ll be using this result for the next two problems, so let’s assign
To find
For the third problem, let’s find the cross product of
Let’s rewrite these in matrix form and start calculating the determinants of the resulting
Practice Questions
Images/mathematical drawings are created with GeoGebra.