Lesson Explainer: Vector Operations in 3D Mathematics

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 𝑖, 𝑗, and 𝑘 or in component form.

Definition: Unit Vectors

A unit vector is a vector of length (magnitude) equal to 1. The unit vectors in the 𝑥, 𝑦, and 𝑧 directions are denoted by 𝑖, 𝑗, and 𝑘 respectively.

Any vector can be written in the form 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘. These can be alternatively represented as (𝑥,𝑦,𝑧) and 𝑥𝑦𝑧.

We will now consider the format of any vector in space whose initial point is at the origin.

In the diagram below, point 𝐴 has coordinates (2,5,3) and vector 𝐴 (which is sometimes denoted as 𝑂𝐴) is the line segment from the origin to point 𝐴.

From the origin, we move 2 units in the 𝑥-direction, 5 units in the 𝑦-direction, and 3 units in the 𝑧-direction such that the vector 𝐴=(2,5,3).

Let us now recall some key definitions about vectors.

Definition: Position Vectors

If point 𝐴 has coordinates (𝑥,𝑦,𝑧), as shown in the diagram, then vector 𝐴=(𝑥,𝑦,𝑧), where the components 𝑥, 𝑦, and 𝑧 are the displacements of point 𝐴 in the 𝑥-, 𝑦-, and 𝑧- 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 𝐴=(𝑥,𝑦,𝑧) and 𝐵=(𝑥,𝑦,𝑧), then 𝐴+𝐵=(𝑥+𝑥,𝑦+𝑦,𝑧+𝑧).

If 𝐴=(𝑥,𝑦,𝑧) and 𝐵=(𝑥,𝑦,𝑧), then 𝐴𝐵=(𝑥𝑥,𝑦𝑦,𝑧𝑧).

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 𝐴=5𝑖8𝑗+6𝑘 and 𝐵=4𝑖3𝑗+13𝑘, find 𝐴𝐵.

Answer

We know that, in order to subtract two vectors in three dimensions, we subtract the corresponding components individually. If 𝐴=(𝑥,𝑦,𝑧) and 𝐵=(𝑥,𝑦,𝑧), then 𝐴𝐵=(𝑥𝑥,𝑦𝑦,𝑧𝑧).

In this question, we need to subtract the 𝑖, 𝑗, and 𝑘 components separately to get 𝐴𝐵=(5,8,6)(4,3,13)=(54,8(3),613)=(9,5,7).

Therefore, 𝐴𝐵=9𝑖5𝑗7𝑘.

Let us now consider how we can add two vectors in three dimensions.

Example 2: Adding Vectors in 3D

Given the two vectors 𝐴=(2,3,0) and 𝐵=(3,3,2), find 𝐴+𝐵.

Answer

We know that, in order to add two vectors in three dimensions, we add the corresponding components individually. If 𝐴=(𝑥,𝑦,𝑧) and 𝐵=(𝑥,𝑦,𝑧), then 𝐴+𝐵=(𝑥+𝑥,𝑦+𝑦,𝑧+𝑧).

This means that 𝐴+𝐵=(2+(3),3+3,0+(2)).

Therefore, 𝐴+𝐵=(5,0,2).

We can extend the rule for adding and subtracting vectors in three dimensions to those in 𝑛-dimensions.

If 𝐴=(𝑎,𝑎,𝑎,,𝑎,𝑎) and 𝐵=(𝑏,𝑏,𝑏,,𝑏,𝑏), then 𝐴+𝐵=(𝑎+𝑏,𝑎+𝑏,𝑎+𝑏,,𝑎+𝑏,𝑎+𝑏), and 𝐴𝐵=(𝑎𝑏,𝑎𝑏,𝑎𝑏,,𝑎𝑏,𝑎𝑏).

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 𝐴=(𝑥,𝑦,𝑧), then 𝑘𝐴=(𝑘𝑥,𝑘𝑦,𝑘𝑧), for all real constants 𝑘.

This can also be extended to the 𝑛-dimensional case. If 𝐴=(𝑎,𝑎,𝑎,,𝑎,𝑎), then 𝑘𝐴=(𝑘𝑎,𝑘𝑎,𝑘𝑎,,𝑘𝑎,𝑘𝑎).

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 𝐴=(6,3,1) by a factor of 6?

Answer

To multiply any vector by a scalar, we multiply each of the individual components by that scalar. If 𝐴=(𝑥,𝑦,𝑧), then 𝑘𝐴=(𝑘𝑥,𝑘𝑦,𝑘𝑧).

In this question, we need to multiply 6, 3, and 1 by 6. We recall that multiplying two negative numbers gives a positive answer: 6×6=36,3×6=18,1×6=6.

So, multiplying (6,3,1) by a factor of 6 gives us the vector (36,18,6).

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 𝐴=(8,9,9) and 𝐵=(6,4,9), find 25𝐴45𝐵.

Answer

To multiply any vector by a scalar, we multiply each of the individual components by that scalar.

Since 𝐴=(8,9,9), then 25𝐴=25(8,9,9)=165,185,185.

As 𝐵=(6,4,9), then 45𝐵=45(6,4,9)=245,165,365.

In order to subtract two vectors in three dimensions, we subtract the corresponding components individually: 25𝐴45𝐵=165,185,185245,165,365=165245,185165,185365=85,25,185.

Therefore, 25𝐴45𝐵=85,25,185.

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 𝐴=(1,1,1) and 𝐵=(1,1,2), determine the vector 𝐶 for which 2𝐶+5𝐴=5𝐵.

Answer

We are told in the question that 2𝐶+5𝐴=5𝐵, so we can begin by rearranging and subtracting 5𝐴 from both sides of the equation. This gives us the equation 2𝐶=5𝐵5𝐴.

Next, we calculate 5𝐴 and 5𝐵. To multiply any vector by a scalar, we multiply each of the individual components by that scalar.

If 𝐴=(1,1,1), then 5𝐴=(5,5,5).

If 𝐵=(1,1,2), then 5𝐵=(5,5,10).

In order to subtract two vectors in three dimensions, we subtract the corresponding components individually.

So, 5𝐵5𝐴=(5,5,10)(5,5,5)=(10,0,15).

As 2𝐶=(10,0,15), we can divide each individual component by 2 in order to calculate vector 𝐶.

Therefore, 𝐶=5,0,152.

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 (𝑥,𝑦) and (𝑥,𝑦), the distance, 𝑑, between them is given by 𝑑=(𝑥𝑥)+(𝑦𝑦).

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 𝐴.

If 𝐴=(𝑥,𝑦,𝑧), then 𝐴=𝑥+𝑦+𝑧.

In our next example, we will calculate the magnitude of vectors in three dimensions.

Example 6: Comparing the Moduli of Vector Expressions

𝑉 and 𝑊 are two vectors, where 𝑉=(1,5,2) and 𝑊=(3,1,1). Comparing 𝑉𝑊 and 𝑉𝑊, 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 𝐴=(𝑥,𝑦,𝑧), then 𝐴=𝑥+𝑦+𝑧.

We are told that 𝑉=(1,5,2).

So, 𝑉=(1)+(5)+(2)=30.

We are also told that 𝑊=(3,1,1).

So, 𝑊=(3)+(1)+(1)=11.

This means that 𝑉𝑊=30112.1606.

In order to subtract two vectors, we subtract the corresponding components individually: 𝑉𝑊=(1,5,2)(3,1,1)=(4,4,3).

So, 𝑉𝑊=(4)+(4)+(3)=41.

So, 416.4031, which is greater than 2.1606.

Therefore, 𝑉𝑊 is larger than 𝑉𝑊.

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 𝐴=3𝑖+𝑗+𝑚𝑘 and that 𝐵 is a unit vector equal to 15𝐴, determine the possible values of 𝑚.

Answer

To multiply any vector by a scalar, we multiply each of the individual components by that scalar.

As 𝐵=15𝐴, then 𝐵=153𝑖+𝑗+𝑚𝑘=35𝑖+15𝑗+𝑚5𝑘.

We are told that 𝐵 is a unit vector, and we know that any unit vector has a magnitude equal to 1, where 𝐵=𝑥+𝑦+𝑧, if 𝐵=(𝑥,𝑦,𝑧): 35+15+𝑚5=1.

Squaring both sides of the equation, 35+15+𝑚5=1,925+125+𝑚25=1.

Multiplying through by 25 and collecting like terms, 10+𝑚=25,𝑚=2510,𝑚=15.

Finding the square root of both sides, 𝑚 could be equal to 15 or 15.

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 𝑥-, 𝑦-, and 𝑧-axes are denoted by 𝑖, 𝑗, and 𝑘 respectively.
  • A vector in 3D space can be written in component form: (𝑥,𝑦,𝑧), or in terms of its fundamental unit vectors: 𝑥𝑖+𝑦𝑗+𝑧𝑘.
  • To add or subtract two vectors, we add or subtract their corresponding components.
    If 𝐴=(𝑥,𝑦,𝑧) and 𝐵=(𝑥,𝑦,𝑧), then 𝐴+𝐵=(𝑥+𝑥,𝑦+𝑦,𝑧+𝑧).
    If 𝐴=(𝑥,𝑦,𝑧) and 𝐵=(𝑥,𝑦,𝑧), then 𝐴𝐵=(𝑥𝑥,𝑦𝑦,𝑧𝑧).
  • To multiply any vector by a scalar, we multiply each of the individual components by that scalar. If 𝐴=(𝑥,𝑦,𝑧), then 𝑘𝐴=(𝑘𝑥,𝑘𝑦,𝑘𝑧).
  • The magnitude of a vector is its length and can be calculated by adapting the Pythagorean theorem in three dimensions. If 𝐴=(𝑥,𝑦,𝑧), then 𝐴=𝑥+𝑦+𝑧.

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