### Video Transcript

In this video, we will learn how to
determine the magnitude of two-dimensional vectors. We will begin by recalling some key
facts about vectors. Any vector has two aspects, its
direction and its magnitude. The magnitude of a vector is its
size or length. There are three main ways that we
can write 2D vectors, as a column vector, in a similar way to a coordinate but with
triangular brackets, or split into ๐ข- and ๐ฃ-components. Each of these three vectors
represent the same thing.

We denote the magnitude of the
vector ๐ฏ by using the absolute value symbol, two parallel vertical lines. We use Pythagorasโs theorem to
calculate it. The magnitude of vector ๐ฏ is equal
to the square root of ๐ squared plus ๐ squared, where ๐ and ๐ are the values
five and two in this case. Our first question involves finding
the magnitude of a vector on a coordinate grid.

The vector ๐ฏ is shown on the grid
of units squares below. Find the value of the magnitude of
๐ฏ.

We know that the magnitude of any
vector is its length. By creating a right triangle on the
grid, we can see that the vector has moved four units to the right and three units
up. The magnitude of vector ๐ฏ can
therefore be found using Pythagorasโs theorem. This states that the length of the
hypotenuse is equal to the sum of the squares of the two shorter sides. The magnitude of ๐ฏ is therefore
equal to the square root of ๐ squared plus ๐ squared.

Whilst it doesnโt matter which
order we substitute the four and the three, we usually do the horizontal component
first. Four squared is equal to 16, and
three squared is equal to nine. The magnitude of vector ๐ฏ is equal
to the square root of 25. As 25 is a square number, we can
calculate this. The square root of 25 is equal to
positive or negative five. As weโre dealing with a length, our
answer must be positive. Therefore, the magnitude of vector
๐ฏ on the grid is five.

We will now look at a couple of
questions where we need to calculate the magnitude of a vector written in different
forms.

What is the magnitude of the vector
five, 12?

We know that for any vector written
in the form ๐, ๐, the magnitude is equal to the square root of ๐ squared plus ๐
squared. As the magnitude is the length of
the vector, this can be shown on a grid. Letโs consider the vector ๐ฏ as
shown. If this vector has moved a distance
๐ in the horizontal direction and ๐ in the vertical direction, we can create a
right triangle. Using Pythagorasโs theorem, the
square of the hypotenuse is equal to ๐ squared plus ๐ squared. This means that the length of the
vector will be equal to the square root of ๐ squared plus ๐ squared.

In this question, the two
components of the vector are five and 12. We can therefore calculate its
magnitude by finding the square root of five squared plus 12 squared. Five squared is equal to 25, and 12
squared is equal to 144. This means that the magnitude of
vector ๐ฏ is the square root of 169. As our answer must be positive, the
magnitude of vector ๐ฏ is 13.

Given that vector ๐ is equal to
negative five ๐ข minus three ๐ฃ, where ๐ข and ๐ฃ are perpendicular unit vectors,
find the magnitude of vector ๐.

We can begin by drawing this on a
grid where ๐ข and ๐ฃ are perpendicular unit vectors. Our vector ๐ moves a distance of
negative five in the ๐ข-direction and a distance of negative three in the
๐ฃ-direction. Vector ๐ can therefore be drawn as
shown. As the magnitude of any vector is
its length, we can calculate this by drawing a right triangle as shown. The magnitude of any vector ๐ฏ can
therefore be calculated using Pythagorasโs theorem, where the magnitude equals the
square root of ๐ squared plus ๐ squared.

Lower case ๐ and ๐ are the ๐ข-
and ๐ฃ-components, respectively. Therefore, the magnitude of vector
๐ is equal to the square root of negative five squared plus negative three
squared. Squaring a negative number gives a
positive answer. So, the square root of negative
five is 25, and the square root of negative three is nine. This means that the magnitude of
vector ๐ is equal to the square root of 34. As 34 is not a square number, we
can leave our answer in surd or radical form. If vector ๐ is equal to negative
five ๐ข minus three ๐ฃ, then its magnitude is equal to the square root of 34.

Our next question will involve
finding the magnitude of a vector between two points.

What is the magnitude of the vector
๐๐ where ๐ด equals 11, three and ๐ต equals seven, three?

The magnitude of a vector is its
size or length. So in this case, we need to find
the distance or length between point ๐ด and point ๐ต. There are several ways of
approaching this problem, we will look at two of them. Our first method will be
graphically, and we will begin by plotting the two coordinates. Point ๐ด has coordinates 11,
three. Point ๐ต has coordinates seven,
three. As both points have the same
๐ฆ-coordinate, the distance from ๐ด to ๐ต will be a horizontal distance. To get from 11 to seven, we need to
subtract four. As the magnitude of any vector must
be positive, then the magnitude of ๐๐ is equal to four.

We could also have calculated the
distance between point ๐ด and point ๐ต using one of our coordinate geometry
formulas. The distance between any two points
is equal to the square root of ๐ฅ one minus ๐ฅ two squared plus ๐ฆ one minus ๐ฆ two
squared, where our two points have coordinates ๐ฅ one, ๐ฆ one and ๐ฅ two, ๐ฆ
two. Substituting in our values gives us
๐ is equal to the square root of 11 minus seven squared plus three minus three
squared.

It doesnโt matter which coordinate
is ๐ฅ one, ๐ฆ one and which one is ๐ฅ two, ๐ฆ two. 11 minus seven is equal to four,
and three minus three is zero. As zero squared is equal to zero,
๐ is equal to the square root of four squared. As our distance must be positive,
this is equal to four. Once again, we have calculated that
the magnitude of the vector ๐๐ is four.

Our final question will involve
finding the magnitude of two separate vectors and their sum.

Consider the vectors ๐ฎ equals two,
three and ๐ฏ equals four, six. What is the magnitude of vector
๐ฎ? What is the magnitude of vector
๐ฏ? What is the magnitude of vector ๐ฎ
plus vector ๐ฏ? In all three questions we need to
give our answer to two decimal places where appropriate.

We recall that the magnitude of any
vector ๐ฐ can be found by square rooting ๐ squared plus ๐ squared, where ๐ and ๐
are the two components of the vector. In vector ๐ฎ, ๐ is equal to two
and ๐ is equal to three. Whereas in vector ๐ฏ, ๐ is equal
to four and ๐ is equal to six. The magnitude of vector ๐ฎ is
therefore equal to the square root of two squared plus three squared. As two squared is equal to four and
three squared is equal to nine, the magnitude of vector ๐ฎ is equal to the square
root of 13. We would often leave this in surd
or radical form. However, in this case, weโre asked
to give our answer to two decimal places. The square root of 13 is equal to
3.605551 and so on.

To round to two decimal places, our
key or deciding number will be the first five. This will round our answer up. The magnitude of vector ๐ฎ to two
decimal places is 3.61. We can repeat this process to
calculate the magnitude of vector ๐ฏ. Four squared is equal to 16, and
six squared is equal to 36. Therefore, the magnitude of vector
๐ฏ is the square root of 52. Typing this into the calculator
gives us 7.211102 and so on. This time our deciding number is a
one. As this is less than five, weโll
round down. The magnitude of vector ๐ฏ is
therefore equal to 7.21.

The final part of our question asks
us to work out the magnitude of ๐ฎ plus ๐ฏ. Our first step here will be to
calculate the vector ๐ฎ plus ๐ฏ. We do this by adding the
corresponding components. Two plus four is equal to six, and
three plus six is equal to nine. We can then calculate the magnitude
of ๐ฎ plus ๐ฏ in the same way. This is equal to the square root of
six squared plus nine squared. Six squared is equal to 36, and
nine squared is equal to 81. Therefore, the magnitude of ๐ฎ plus
๐ฏ is equal to the square root of 117. Typing this into the calculator
gives us 10.816653. The deciding number here is a six,
and anything five or greater means that we round up. The magnitude of ๐ฎ plus ๐ฏ is
equal to 10.82.

When looking at our three answers,
you might think youโve spotted a pattern, as 3.61 plus 7.21 is equal to 10.82. This suggests that the magnitude of
๐ฎ plus ๐ฏ is equal to the magnitude of ๐ฎ plus the magnitude of ๐ฏ. This, however, is not normally the
case. The only reason this works in this
question is that vector ๐ฏ is actually a multiple of vector ๐ฎ. Two multiplied by two is equal to
four. And three multiplied by two is
equal to six.

Therefore, vector ๐ฏ is actually
two lots or two multiplied by vector ๐ฎ. This in turn means that the
magnitude of vector ๐ฏ is twice the magnitude of vector ๐ฎ. Root 52 is equal to two root
13. The magnitude of ๐ฎ plus ๐ฏ in this
question becomes three times the magnitude of vector ๐ฎ. It is important to note, however,
as previously mentioned, this will not hold for the majority of vector
questions.

We will now summarize the key
points from this video. The magnitude of a vector is its
length. We can calculate the magnitude of
any vector in two dimensions using Pythagorasโs theorem. The magnitude of vector ๐ฏ is equal
to the square root of ๐ squared plus ๐ squared, where ๐ and ๐ are the two
components of the vector. Whilst our answer is usually
written as a surd or radical, we can calculate the decimal value.

Finally, we found out that in the
majority of cases, the magnitude of vector ๐ฎ plus vector ๐ฏ is not equal to the
magnitude of vector ๐ฎ plus the magnitude of vector ๐ฏ. Everything we have used in this
video can also be applied to three-dimensional vectors. These would be written in the form
๐, ๐, ๐, the column vector ๐, ๐, ๐, or ๐๐ข plus ๐๐ฃ plus ๐๐ค.