In this explainer, we will learn how to do operations on vectors in 3D, such as addition, subtraction, and scalar multiplication.
The vector operations of addition, subtraction, and scalar multiplication work in the same way in three or more dimensions as they do in two dimensions. We will begin by recalling what a vector written in three dimensions looks like.
A vector drawn in three dimensions has a tail (initial point) and head (terminal point). The direction of the vector is denoted by an arrow and the length of the vector is known as its magnitude. We can write a vector in terms of its unit vectors , , and or in component form.
Definition: Unit Vectors
A unit vector is a vector of length (magnitude) equal to 1. The unit vectors in the , , and directions are denoted by , , and respectively.
Any vector can be written in the form + + . These can be alternatively represented as and .
We will now consider the format of any vector in space whose initial point is at the origin.
In the diagram below, point has coordinates and vector (which is sometimes denoted as ) is the line segment from the origin to point .
From the origin, we move 2 units in the -direction, 5 units in the -direction, and 3 units in the -direction such that the vector .
Let us now recall some key definitions about vectors.
Definition: Position Vectors
If point has coordinates , as shown in the diagram, then vector , where the components , , and are the displacements of point in the -, -, and - direction from the origin, is called a position vector.
Definition: Adding and Subtracting Vectors
We can add or subtract any two vectors by adding or subtracting their corresponding components.
If and , then .
If and , then .
In our first example, we will demonstrate how to subtract one vector from another when they are both given in terms of their unit vectors.
Example 1: Subtracting Vectors in 3D
If and , find .
Answer
We know that, in order to subtract two vectors in three dimensions, we subtract the corresponding components individually. If and , then .
In this question, we need to subtract the , , and components separately to get
Therefore, .
Let us now consider how we can add two vectors in three dimensions.
Example 2: Adding Vectors in 3D
Given the two vectors and , find .
Answer
We know that, in order to add two vectors in three dimensions, we add the corresponding components individually. If and , then .
This means that .
Therefore, .
We can extend the rule for adding and subtracting vectors in three dimensions to those in -dimensions.
If and , then , and .
Definition: Multiplying a Vector by a Scalar
To multiply any vector by a scalar, we multiply each of the individual components by that scalar.
If , then , for all real constants .
This can also be extended to the -dimensional case. If , then .
In our next example, we will demonstrate how we can multiply a vector by a scalar quantity.
Example 3: Scaling a 3D Vector
What is the vector that results from scaling the vector by a factor of ?
Answer
To multiply any vector by a scalar, we multiply each of the individual components by that scalar. If , then .
In this question, we need to multiply , , and by . We recall that multiplying two negative numbers gives a positive answer:
So, multiplying by a factor of gives us the vector .
In the fourth example, we will combine the multiplication of a vector by a scalar with subtraction of vectors.
Example 4: Subtracting Scalar Multiples of Vectors
If and , find .
Answer
To multiply any vector by a scalar, we multiply each of the individual components by that scalar.
Since , then
As , then
In order to subtract two vectors in three dimensions, we subtract the corresponding components individually:
Therefore, .
In our next example, we will find the missing vector in a vector expression.
Example 5: Finding an Unknown Vector Given a Vector Expression
If and , determine the vector for which .
Answer
We are told in the question that , so we can begin by rearranging and subtracting from both sides of the equation. This gives us the equation .
Next, we calculate and . To multiply any vector by a scalar, we multiply each of the individual components by that scalar.
If , then .
If , then .
In order to subtract two vectors in three dimensions, we subtract the corresponding components individually.
So,
As , we can divide each individual component by 2 in order to calculate vector .
Therefore, .
When given two points in space, we can apply the distance formula to find the distance between them. This is a variant of the Pythagorean theorem. Given two points and , the distance, , between them is given by
This can be generalized even further to give us the distance between a point in three-dimensional space and the origin. In vector terms, this means that we can find the length of a vector, which we call the magnitude of the vector.
Definition: Magnitude of a Vector
The magnitude of a vector tells us its length and is denoted by .
If , then .
In our next example, we will calculate the magnitude of vectors in three dimensions.
Example 6: Comparing the Moduli of Vector Expressions
and are two vectors, where and . Comparing and , which quantity is larger?
Answer
In order to calculate the magnitude of any vector, we calculate the square root of the sum of the squares of the individual components. If , then .
We are told that .
So, .
We are also told that .
So, .
This means that .
In order to subtract two vectors, we subtract the corresponding components individually:
So,
So, , which is greater than 2.1606.
Therefore, is larger than .
In our last example, we demonstrated that the magnitude of the difference of two vectors is not equal to the difference between their respective magnitudes. It is important to realize that while we can find the sum or difference of two or more vectors fairly easily, we cannot apply a similar concept to the sum or difference of their magnitudes.
In our final example, we will calculate the possible missing values in a vector problem.
Example 7: Solving a Vector Problem Involving Unit Vectors
Given that and that is a unit vector equal to , determine the possible values of .
Answer
To multiply any vector by a scalar, we multiply each of the individual components by that scalar.
As , then
We are told that is a unit vector, and we know that any unit vector has a magnitude equal to 1, where , if :
Squaring both sides of the equation,
Multiplying through by 25 and collecting like terms,
Finding the square root of both sides, could be equal to or .
We will finish this explainer by recapping some of the key points.
Key Points
- A unit vector has a magnitude of 1, and the unit vectors parallel to the -, -, and -axes are denoted by , , and respectively.
- A vector in 3D space can be written in component form: , or in terms of its fundamental unit vectors: .
- To add or subtract two vectors, we add or subtract their corresponding components.
If and , then .
If and , then . - To multiply any vector by a scalar, we multiply each of the individual components by that scalar. If , then .
- The magnitude of a vector is its length and can be calculated by adapting the Pythagorean theorem in three dimensions. If , then .
