Lesson Video: Properties of Operations on Vectors Mathematics

Video Transcript

In this video, we will learn how to use the properties of addition and multiplication on vectors. We begin by recalling that a vector is a quantity with both a magnitude and a direction. And we can represent a vector in a suitable space by a directed line segment with a specific length. In this video, we will only consider vectors in two dimensions.

In two dimensions, we can represent the magnitude and direction of a vector in terms of horizontal and vertical change as shown. If the vector 𝐯 has a horizontal component 𝑎 and a vertical component 𝑏, we can think of this as a displacement of 𝑎 units horizontally and a displacement of 𝑏 units vertically. We can use this idea to add two vectors together by considering their components.

Graphically, the sum of two vectors 𝐮 and 𝐯 is their combined displacement. We can therefore sketch the terminal point of the first vector as the initial point of the second vector. Then the sum of the vectors has the initial point of the first vector and the terminal point of the second vector as shown. Since the vector 𝐮 plus 𝐯 represents the displacement of both vector 𝐮 and vector 𝐯, it will have a horizontal component equal to the sum of the horizontal components of vector 𝐮 and vector 𝐯 and a vertical component equal to the sum of the vertical components of vector 𝐮 and vector 𝐯.

This can be written more formally as follows. For any two vectors in two dimensions 𝐮 with components 𝑢 sub one and 𝑢 sub two and 𝐯 with components 𝑣 sub one and 𝑣 sub two, then 𝐮 plus 𝐯 has components 𝑢 sub one plus 𝑣 sub one and 𝑢 sub two plus 𝑣 sub two. Since the sum of any two vectors in two dimensions is also a two-dimensional vector, we can say that vector addition in two dimensions is closed. This is sometimes referred to as the closure property of vector addition. Whilst this idea does extend to higher dimensions, in this video, we will only be working in two dimensions. We can also define scalar multiplication on a vector as scalar multiplication of its components.

For any vector 𝐮 with components 𝑢 sub one and 𝑢 sub two and scalar 𝑘, then 𝑘 multiplied by vector 𝐮 has components 𝑘𝑢 sub one and 𝑘𝑢 sub two. Graphically, scalar multiplication of a vector by scalar 𝑘 is a dilation or enlargement of the vector by a factor 𝑘. We will now look at an example of how to use these definitions to answer a question involving a property of vector addition.

Complete the following: the vector one, nine plus the vector five, two is equal to the vector five, two plus what.

We begin here by simplifying the left-hand side of our equation. We begin by recalling that to find the sum of a pair of vectors, we simply add their corresponding components. In this question, to add the vectors one, nine and five, two, we add one and five and then separately nine and two. This means that the left-hand side of our equation is equal to the vector six, 11. If we let the missing vector on the right-hand side have components 𝑥 and 𝑦, we can simplify the right-hand side as shown. The vector five, two plus the vector 𝑥, 𝑦 gives us the vector five plus 𝑥, two plus 𝑦.

We can now equate the two sides of our equation. Six, 11 is equal to five plus 𝑥, two plus 𝑦. For the two vectors to be equal, we know that their corresponding components must be equal. This gives us two equations we need to solve: six is equal to five plus 𝑥 and 11 is equal to two plus 𝑦. Subtracting five from both sides of our first equation, we see that 𝑥 is equal to one. And subtracting two from both sides of our second equation, we see that 𝑦 is equal to nine. The missing vector is therefore equal to one, nine. The vector one, nine plus the vector five, two is equal to the vector five, two plus the vector one, nine.

This question demonstrates the commutativity of vector addition, which we’ll now summarize.

For any two vectors in two dimensions 𝐮 and 𝐯, vector 𝐮 plus vector 𝐯 is equal to vector 𝐯 plus vector 𝐮. A graphical interpretation of the property is shown in the diagram. If the two vectors 𝐮 and 𝐯 are nonzero, then we can sketch these vectors as the sides of a parallelogram. This means that the vector of the diagonal of the parallelogram can be represented as both 𝐮 plus 𝐯 and 𝐯 plus 𝐮. Therefore, these expressions must be equal. In the case where one of the vectors is the zero vector, this leads us to the additive identity property. This is one of many properties of vector addition and scalar multiplication in two dimensions. Whilst we’ll not prove these properties in this video, we will list them now.

For any vectors 𝐮, 𝐯, and 𝐰 and scalars 𝑚 and 𝑛, we will consider the following: firstly, five properties of vector addition. We have already seen that vector 𝐮 plus vector 𝐯 is equal to vector 𝐯 plus vector 𝐮. This is the commutative property. Secondly, we have vector 𝐮 plus vector 𝐯 plus vector 𝐰 is equal to vector 𝐮 plus vector 𝐯 plus vector 𝐰. This is known as the associative property and means that when we add three vectors, it doesn’t matter which two vectors we add first. Next, we have vector 𝐮 plus the zero vector is equal to vector 𝐮. This is known as the additive identity property and means that if we add the zero vector to any vector, the vector remains unchanged.

Next, we have the additive inverse property, which states that vector 𝐮 plus negative vector 𝐮 is equal to the zero vector. Adding any vector to its inverse always gives us the zero vector. Finally, we have the elimination property. This states that if 𝐮 plus 𝐯 is equal to 𝐮 plus 𝐰, then 𝐯 is equal to 𝐰. These are the five properties of vector addition.

We also need to consider five properties of scalar multiplication of vectors. There are two forms of the distributive property. There is the multiplicative identity property, the associative property, and, once again, the elimination property. Multiplying the scalar 𝑛 by the vector sum 𝐮 plus 𝐯 gives us 𝑛𝐮 plus 𝑛𝐯. And multiplying the sum of two scalars 𝑛 and 𝑚 by the vector 𝐮 gives us 𝑛𝐮 plus 𝑚𝐮. Multiplying any vector 𝐮 by the scalar one gives us the vector 𝐮. This is known as the multiplicative identity property. The associative property states that 𝑛𝑚 multiplied by the vector 𝐮 is equal to 𝑛 multiplied by 𝑚𝐮. Finally, the elimination property states that if 𝑛𝐮 is equal to 𝑛𝐯, then vector 𝐮 must be equal to vector 𝐯.

All 10 of these properties are true for vectors in dimensions higher than two and as previously mentioned approvable algebraically. In the remainder of this video, we will look at examples of how we can use these properties to evaluate expressions involving vectors.

Given that vector 𝐚 equals one, five and vector 𝐛 equals six, two, find 𝐚 plus 𝐛 plus negative 𝐚.

We can answer this question directly using the properties of vector addition. Firstly, using the commutative property, which states that vector 𝐮 plus vector 𝐯 is equal to vector 𝐯 plus vector 𝐮, we can rewrite our expression 𝐚 plus 𝐛 plus negative 𝐚 as 𝐚 plus negative 𝐚 plus 𝐛. Next, we’ll use the additive inverse property, which states that vector 𝐮 plus negative vector 𝐮 is equal to the zero vector. Applying this to our expression, vector 𝐚 plus negative vector 𝐚 is equal to the zero vector. So we’re left with the zero vector plus vector 𝐛.

Finally, we’ll use the additive identity property, which states that vector 𝐮 plus the zero vector is equal to vector 𝐮. This means that, in our question, the zero vector plus vector 𝐛 is simply equal to vector 𝐛. We are told in the question that vector 𝐛 is equal to six, two. This means that 𝐚 plus 𝐛 plus negative 𝐚 is also equal to six, two. A second method here would be to simply work with the components of vector 𝐚 and vector 𝐛. We need to add the vectors one, five and six, two and then add the negative of the vector one, five. We can distribute the negative over the vector by multiplying all of its components by negative one. Therefore, the third vector becomes negative one, negative five.

We can now just add the three vectors by finding the sum of their corresponding components. We begin by adding one, six, and negative one. This is equal to six. We then add the 𝑦-components of five, two, and negative five, giving us two. This confirms the answer we got using the properties of vector addition. Vector 𝐚 plus vector 𝐛 plus negative vector 𝐚 is equal to six, two.

We will now consider one final example.

Complete the following: two multiplied by the vector two, five plus the vector five, one is equal to what plus the vector 10, two.

We begin this question by simplifying the left-hand side of the equation. Firstly, we use the fact that scalar multiplication is distributive over vector addition. This means that the left-hand side becomes two multiplied by the vector two, five plus two multiplied by the vector five, one. We can then evaluate the scalar multiplication. Two multiplied by the vector two, five is equal to the vector two multiplied by two, two multiplied by five. This is equal to four, 10. Likewise, multiplying the vector five, one by the scalar two gives us the vector 10, two. We can then equate this to the right-hand side of our equation and let the components of the unknown vector be 𝑥 and 𝑦.

Next, we can consider the elimination property of vector addition, which states that if vector 𝐮 plus vector 𝐯 is equal to vector 𝐮 plus vector 𝐰, then vector 𝐯 is equal to vector 𝐰. The vector 10, two appears in the sum on both sides of our equation. This means that the other vectors on each side must also be equal. The vector four, 10 is equal to the vector 𝑥, 𝑦. We can therefore conclude that the missing vector is four, 10.

We will complete this video by summarizing the key points. We saw in this video that we can use the properties of vector addition and scalar multiplication to simplify expressions using vectors. The five properties of vector addition are as shown. These are known as the commutative property, associative property, additive identity property, additive inverse property, and elimination property, respectively. The five properties of scalar multiplication of vectors are as shown. The first two here are examples of the distributive property. The third is the multiplicative identity property, followed by the associative property and, once again, the elimination property.

We can prove that all these properties hold by considering the components of the vectors. Whilst we’ve only considered the properties for vectors in two dimensions in this video, all of these properties extend to vectors in higher dimensions.