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.
