Lesson Explainer: Scalar Multiplication and Unit Vectors Mathematics

In this explainer, we will learn how to multiply a vector by a scalar and how to find a unit vector in the direction of any given vector by dividing the vector by a scalar.

A vector is a quantity that has both a magnitude and a direction. The combination of these features makes the vector a versatile tool that has many applications. Some examples include describing the position of a point in space relative to another point and expressing the velocity of a moving body.

Vectors are often expressed by relating them to a set of coordinates and, hence, have more than one component. A 3D vector ⃑𝑉 is often expressed in terms of the Cartesian coordinates π‘₯, 𝑦, and 𝑧, ⃑𝑉=(π‘₯,𝑦,𝑧), or using unit vectors, ⃑𝑉=π‘₯⃑𝑖+𝑦⃑𝑗+π‘§βƒ‘π‘˜.

In contrast to this, a scalar is a quantity that has a magnitude but no direction. A scalar can often be represented by a single number, with no extra information needed.

Multiplying two scalars together is simple, but in this explainer we are going to explore scalar multiplication of a vector. As the name suggests, this involves multiplying a vector by a scalar.

Consider a particle moving with a velocity ⃑𝑉. If the particle continues to move in the same direction but its velocity doubles in magnitude, how can we describe this? The simple answer is 2⃑𝑉. Since we have multiplied a scalar by a vector, this is an example of scalar multiplication.

Let’s see how this relates to the vector components.

Definition: Scalar Multiplication

Multiplying a scalar by a vector is known as scalar multiplication.

Consider the vector ⃑𝑉 and the scalar π‘˜: ⃑𝑉=(π‘₯,𝑦,𝑧).

When performing a scalar multiplication, the scalar can be distributed across the vector components: π‘˜βƒ‘π‘‰=π‘˜(π‘₯,𝑦,𝑧)=(π‘˜π‘₯,π‘˜π‘¦,π‘˜π‘§).

The result of scalar multiplication is also a vector, where the magnitude of the original vector is multiplied by the value represented by the scalar.

We will go through a simple example before going into some more intuitive interpretations of scalar multiplication.

Example 1: Multiplying a 2D Vector by a Scalar

Given that ⃑𝐴=(βˆ’1,βˆ’8), find 3⃑𝐴.

Answer

For this question, we have been given a 2D vector and asked to perform a multiplication by the scalar quantity of 3.

In order to do this, we can simply multiply both the π‘₯ and the 𝑦 component of the vector by the given scalar: 3⃑𝐴=3(βˆ’1,βˆ’8)=(3(βˆ’1),3(βˆ’8))=(βˆ’3,βˆ’24).

With some simplification, we find our result is a vector with an π‘₯ component of βˆ’3 and a 𝑦 component of βˆ’24.

Note that the vector 3⃑𝐴 has the same direction as ⃑𝐴, but its magnitude is now larger, having been scaled by a factor of 3.

The question above gives us a numerical example of scalar multiplication by a positive number. Since vectors are often represented as arrows, let’s take a look at some visual examples.

As we have seen, multiplying a vector by a positive scalar does not change the direction of the resulting vector but scales its magnitude.

Let’s consider the visual representation of a vector, ⃑𝑉, under various forms of scalar multiplication.

Consider section (A) of the diagram. Multiplying vector ⃑𝑉 by 1 would give us the same result as the original vector, and its magnitude would be unchanged; however, multiplying by 3 would increase the magnitude by a factor of 3.

Section (B) of the diagram shows that it is important to understand that multiplying a vector by a negative number does change its direction. Let’s consider the simplest case, where vector ⃑𝑉 is multiplied by βˆ’1. The magnitude of the vector will remain unchanged but its direction will be reversed since all of the components will change sign: ⃑𝑉=(π‘₯,𝑦)βˆ’βƒ‘π‘‰=(βˆ’π‘₯,βˆ’π‘¦).

Multiplying by a negative number other than βˆ’1 will reverse the direction and also scale the magnitude of the vector.

Finally, for section (C) of the diagram, we note that scalar multiplication is not confined to integers nor to numbers that are greater than 1. If the magnitude of the scalar is less than 1, the scalar multiplication will result in a vector with a smaller magnitude (or a shorter length) than the original.

Example 2: Multiplying a 2D Vector by a Scalar Graphically

⃑𝐴 is represented by the following graph.

Which of the following graphs represents βˆ’2⃑𝐴?

Answer

Let’s first consider a less formal solution to this problem.

The question gives us a vector ⃑𝐴 and asks us to perform a scalar multiplication by the factor of βˆ’2. Since we are dealing with a negative scalar, we know that this operation will reverse the direction of our original vector.

Vector ⃑𝐴 has its start point at the origin and its endpoint in the top right quadrant. If the resulting vector were to also start at the origin (as seen in all of the options), we know that its endpoint must lie in the bottom left quadrant, since the direction is reversed.

The only graph that matches this description is option (a), and so it seems this is the correct answer.

We may also consider the magnitude of our scalar. Since |βˆ’2|=2, the vector that results from our scalar multiplication will have double the length of the original. Visually inspecting option (a), this seems to be the case!

To further verify our solution, we may take a more formal approach by first defining vector ⃑𝐴. From our graph, we see that the start point of ⃑𝐴 is the origin, which has coordinates (0,0), and the endpoint lies at (1,1). It therefore follows that ⃑𝐴=(1,1)βˆ’(0,0)=(1,1).

Now that we have defined vector ⃑𝐴, we can perform a scalar multiplication as follows: βˆ’2⃑𝐴=βˆ’2(1,1)=(βˆ’2(1),βˆ’2(1))=(βˆ’2,βˆ’2).

As mentioned, all of the multiple-choice options we have involve a vector that starts at the origin. If the vector (βˆ’2,βˆ’2), starts at the origin, its endpoint will be at the coordinates (βˆ’2,βˆ’2).

The graph that matches this description is option (a), and hence we have confirmed this is the correct answer.

Another important concept that can be related to scalar multiplication is that of the unit vector. We are familiar with the unit vectors ⃑𝑖, ⃑𝑗, and βƒ‘π‘˜. These are vectors of magnitude 1 in the π‘₯-, 𝑦-, and 𝑧-directions respectively.

In actual fact, any vector of length 1 can be considered a unit vector! We can therefore define a unit vector in any given direction or for any set of coordinates. Let’s formally generalize this concept.

Definition: Unit Vectors

A unit vector is a vector that has a magnitude of length 1.

We can find the unit vector, denoted ⃑𝑣, in the direction of ⃑𝑉 by dividing a vector by its magnitude: ⃑𝑣=⃑𝑉‖‖⃑𝑉‖‖.

Recall that the magnitude of the vector ⃑𝑉=(π‘₯,𝑦,𝑧) is given by ‖‖⃑𝑉‖‖=√π‘₯+𝑦+𝑧.

Given the above definition of a unit vector, we should note that an equivalent mathematical statement is ⃑𝑣=⃑𝑉‖‖⃑𝑉‖‖.

Note that the magnitude ‖‖⃑𝑉‖‖ has no sense of direction and is therefore a scalar. This means that 1‖‖⃑𝑉‖‖ is also a scalar.

Looking back at our formula, we can now recognize that it involves the multiplication of a scalar 1‖‖⃑𝑉‖‖ and a vector ⃑𝑉. The process of finding a unit vector in a given direction can hence be thought of as a β€œspecial case” of scalar multiplication!

Instead of multiplying a vector by a given scalar, we find the scalar that will give our result the desired magnitude. In the case of ⃑𝑉=(π‘₯,𝑦,𝑧), our calculation might look like this: ⃑𝑣=1‖‖⃑𝑉‖‖⃑𝑉=1√π‘₯+𝑦+𝑧(π‘₯,𝑦,𝑧).

Since scalar multiplication by a positive number does not change the direction of a vector, the result will be a new vector that points in the same direction of ⃑𝑉 but has a magnitude of 1. This is the definition of ⃑𝑉.

As a quick side note, while it is standard to use a circumflex accent (or β€œhat”) such as 𝑉 to denote a unit vector, you will commonly see ̂𝑖, ̂𝑗, andΒ Μ‚π‘˜Β without this notation. Since these three unit vectors are so commonly used, they are often represented as just ⃑𝑖, ⃑𝑗, andΒ  βƒ‘π‘˜.

Let’s look at an example of finding a unit vector.

Example 3: Finding a Unit Vector in terms of the Original Vector

Find the unit vector in the same direction as the vector βƒ‘π‘–βˆ’βƒ‘π‘—βˆ’βƒ‘π‘˜.

Answer

This question has given us the vector βƒ‘π‘–βˆ’βƒ‘π‘—βˆ’βƒ‘π‘˜ and asked us to find the unit vector in the same direction. This will be a vector of magnitude 1. Note that we have been given our vector in terms of the unit vectors ⃑𝑖, ⃑𝑗, and βƒ‘π‘˜ but this has no effect on the method we will use.

Our first step is to find the magnitude of βƒ‘π‘–βˆ’βƒ‘π‘—βˆ’βƒ‘π‘˜. Note that while the individual components of this vector in the π‘₯-, 𝑦-, and 𝑧-directions are all equal to 1, this does not mean the vector itself has a magnitude of 1: β€–β€–βƒ‘π‘–βˆ’βƒ‘π‘—βˆ’βƒ‘π‘˜β€–β€–=1+(βˆ’1)+(βˆ’1)=√1+1+1=√3.

Now recall that, to find the unit vector of some vector ⃑𝑉, we divide it by its magnitude: ⃑𝑣=⃑𝑉‖‖⃑𝑉‖‖=1‖‖⃑𝑉‖‖⃑𝑉.

In order to find the unit vector for the given question, we can therefore perform the following calculation: 1√3ο€»βƒ‘π‘–βˆ’βƒ‘π‘—βˆ’βƒ‘π‘˜ο‡=√33ο€»βƒ‘π‘–βˆ’βƒ‘π‘—βˆ’βƒ‘π‘˜ο‡.

In the above steps, we have performed the common technique of rationalizing the denominator to reach an answer. This formof answer is perfectly valid since it expresses the unit vector as a multiple of the original vector.

An equivalent expression can be obtained by multiplying the individual components of the vector by the scalar: =√33βƒ‘π‘–βˆ’βˆš33βƒ‘π‘—βˆ’βˆš33βƒ‘π‘˜.

For our final example, we will be combining our skills in scalar multiplication along with other vector operations in order to solve a unit vector problem.

Example 4: Combining Scalar Multiplication, Vector Operations, and Unit Vector Calculations

Given ⃑𝐴=(2,0,βˆ’2) and ⃑𝐡=(1,βˆ’1,1), determine the unit vector in the direction 2βƒ‘π΅βˆ’βƒ‘π΄.

Answer

This question asks us to find a unit vector in the direction of 2βƒ‘π΅βˆ’βƒ‘π΄, and in order to do this, we will need the magnitude of the vector 2βƒ‘π΅βˆ’βƒ‘π΄.

To find this magnitude, let’s first find the components of 2βƒ‘π΅βˆ’βƒ‘π΄. This will require a combination of scalar multiplication and vector subtraction: 2βƒ‘π΅βˆ’βƒ‘π΄=2(1,βˆ’1,1)βˆ’(2,0,βˆ’2).

We can form an equation using the vectors ⃑𝐴 and ⃑𝐡 given in the question. Notice that the first term on the right-hand side is a scalar multiplication. The scalar of 2 can be distributed among the components of ⃑𝐡. Following this, we will be able to perform the vector subtraction: 2βƒ‘π΅βˆ’βƒ‘π΄=(2,βˆ’2,2)βˆ’(2,0,βˆ’2)=(0,βˆ’2,4).

Now that we have the components of 2βƒ‘π΅βˆ’βƒ‘π΄, we can find its magnitude: β€–β€–2βƒ‘π΅βˆ’βƒ‘π΄β€–β€–=0+(βˆ’2)+4=√0+4+16=√20=2√5.

Note that, for our last step, we have simplified our magnitude by taking a factor of 2 outside of the radical.

Now that we have found the components of 2βƒ‘π΅βˆ’βƒ‘π΄ and its magnitude, we can find the unit vector in the direction of 2βƒ‘π΅βˆ’βƒ‘π΄ by applying the following rule: ⃑𝑣=⃑𝑉‖‖⃑𝑉‖‖=1‖‖⃑𝑉‖‖⃑𝑉.

In order to find our unit vector, we divide 2βƒ‘π΅βˆ’βƒ‘π΄ by its magnitude. Another way of expressing this is to multiply the reciprocal of the magnitude by the original vector: 1β€–β€–2βƒ‘π΅βˆ’βƒ‘π΄β€–β€–ο€Ί2βƒ‘π΅βˆ’βƒ‘π΄ο†=12√5(0,βˆ’2,4)=√510(0,βˆ’2,4)=ο€Ώ0,βˆ’2√510,4√510=ο€Ώ0,βˆ’βˆš55,2√55.

After a few simplification steps, we reach our answer.

As a final note, it is possible to perform a quick check by confirming that our answer is a vector of magnitude 1: ο„‘ο„£ο„£ο„ 0+ο€Ώβˆ’βˆš55+ο€Ώ2√55=ο„ž525+2025=ο„ž2525=1.

Of course, this step is not necessary in this case, but it can sometimes be useful when dealing with longer calculations.

Let’s summarize some of the key points relating to scalar multiplication and unit vectors.

Key Points

  • Multiplying a scalar by a vector is known as scalar multiplication.
  • When performing a scalar multiplication, the scalar can be distributed across the vector components: π‘˜βƒ‘π‘‰=π‘˜(π‘₯,𝑦,𝑧)=(π‘˜π‘₯,π‘˜π‘¦,π‘˜π‘§).
  • The result of scalar multiplication is also a vector, where the magnitude of the original vector is multiplied by the value represented by the scalar.
  • When the scalar is a positive number, the resulting vector is in the same direction as the original. When the scalar is negative, the resulting vector is in the opposite direction to the original.
  • A unit vector is a vector that has a magnitude of length 1.
  • We can find the unit vector, denoted ⃑𝑣, in the direction of ⃑𝑉 by dividing the original vector by its magnitude: ⃑𝑣=⃑𝑉‖‖⃑𝑉‖‖=1‖‖⃑𝑉‖‖⃑𝑉.

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