### Video Transcript

In this video, we will learn how to
find the magnitude of a position vector in space. We will begin by recalling what we
mean by a 3D vector.

A three-dimensional vector has an
๐ข-, ๐ฃ-, and ๐ค-component. If we consider the point ๐ with
coordinates two, three, five, we can write the vector ๐๐ in numerous ways. Firstly, we can consider the ๐ข-,
๐ฃ-, and ๐ค-components. The vector ๐๐ is equal to two ๐ข
plus three ๐ฃ plus five ๐ค. Vectors are also sometimes written
similar to coordinates. With triangular brackets, we have
two, three, five. A third way of writing the same
vector is as a column inside parentheses, with the ๐ข-component followed by the
๐ฃ-component and then the ๐ค-component.

The magnitude of any vector is the
distance between two points in three-dimensional space. If we consider a vector ๐ written
in the general form ๐ฅ๐ข plus ๐ฆ๐ฃ plus ๐ง๐ค, then the magnitude of vector ๐ can be
calculated using an application of the Pythagorean theorem. The magnitude of vector ๐ denoted
by two vertical lines is equal to the square root of ๐ฅ squared plus ๐ฆ squared plus
๐ง squared. We find the sum of the squares of
the ๐ข-, ๐ฃ-, and ๐ค-components and then square root the answer.

As well as seeing our unit vectors
๐ข, ๐ฃ, and ๐ค written with a hat, you may also have seen them underlined. These notations for handwritten
vectors will vary from location to location. Generally when typed, either in a
text book or on the internet, vectors will appear in bold. We will now look at some questions
where we need to calculate the magnitude of a vector.

If vector ๐ is equal to two,
negative five, two, find the magnitude of vector ๐.

The ๐ข-, ๐ฃ-, and ๐ค-components of
vector ๐ are two, negative five, and two, respectively. Therefore, vector ๐ could be
rewritten as two ๐ข minus five ๐ฃ plus two ๐ค. We recall that the magnitude of any
vector could be calculated by square rooting ๐ฅ squared plus ๐ฆ squared plus ๐ง
squared, where ๐ฅ, ๐ฆ, and ๐ง are the ๐ข-, ๐ฃ-, and ๐ค-components, respectively. The magnitude of vector ๐ is,
therefore, equal to the square root of two squared plus negative five squared plus
two squared.

Two squared is equal to four. Squaring a negative number gives a
positive answer. Therefore, negative five squared is
25. The magnitude of ๐ is equal to the
square root of four plus 25 plus four. This is equal to the square root of
33. Whilst we could work out this
answer on the calculator, as a general rule, we will leave our answers as radicals
or surds. The magnitude of vector ๐ is the
square root of 33.

In our next question, the vector
will be written in a different format.

If vector ๐ is equal to two ๐ข
plus three ๐ฃ minus ๐ค, find the magnitude of vector ๐.

For any vector written in the form
๐ฅ๐ข plus ๐ฆ๐ฃ plus ๐ง๐ค, the magnitude of the vector is equal to the square root of
๐ฅ squared plus ๐ฆ squared plus ๐ง squared. The ๐ข-component of our vector is
equal to two, the ๐ฃ-component is equal to three, and the ๐ค-component is equal to
negative one. This means that the magnitude of
vector ๐ is equal to the square root of two squared plus three squared plus
negative one squared.

Two squared is equal to four. Three squared is equal to nine. Squaring a negative number gives us
a positive answer. Therefore, negative one squared is
one. As four plus nine plus one equals
14, the magnitude of vector ๐ is the square root of 14.

In our next question, we will be
given the magnitude and need to calculate one of the components of the vector.

If vector ๐ is equal to ๐๐ข plus
๐ฃ minus ๐ค and the magnitude of vector ๐ is equal to the square root of six, find
all the possible values of ๐.

Before starting this question, it
is worth noting that the wording says find all possible values of ๐. This suggests there will be more
than one correct answer. We are given two pieces of
information. We are told vector ๐ is equal to
๐๐ข plus ๐ฃ minus ๐ค and the magnitude of vector ๐ is the square root of six. We know that for any vector written
in the form ๐ฅ๐ข plus ๐ฆ๐ฃ plus ๐ง๐ค, then its magnitude is equal to the square root
of ๐ฅ squared plus ๐ฆ squared plus ๐ง squared. In this question, the square root
of six is equal to the square root of ๐ squared plus one squared plus negative one
squared. This is because the ๐ข-, ๐ฃ-, and
๐ค-components are ๐, one, and negative one, respectively.

We can begin to solve this equation
by squaring both sides. As squaring is the inverse or
opposite of square rooting, the square root of six squared is equal to six. In the same way, the right-hand
side becomes ๐ squared plus one squared plus negative one squared. Both one squared and negative one
squared are equal to one. Therefore, this simplifies to six
is equal to ๐ squared minus two. We can then subtract two from both
sides of this equation so that ๐ squared is equal to four.

Our final step is to square root
both sides. The square root of ๐ squared is
๐. The square root of four is equal to
two. But we must take the positive or
negative of this. Therefore, ๐ is equal to positive
or negative two. The possible values of ๐ such that
the magnitude of vector ๐ is the square root of six are two and negative two. This is because when we square both
of these, we get an answer of four.

Our next few questions will also
involve the addition and subtraction of vectors.

Given that vector ๐ plus vector ๐
is equal to negative two, four, three and vector ๐ is equal to three, five, three,
determine the magnitude of vector ๐.

We recall that when adding two
vectors, we simply add the ๐ข-, ๐ฃ-, and ๐ค-components separately. The vector ๐ plus ๐ is equal to
the vector ๐ plus the vector ๐. If we let vector ๐ have ๐ข-, ๐ฃ-,
and ๐ค-components ๐ฅ, ๐ฆ, and ๐ง, respectively, then negative two, four, three is
equal to three, five, three plus ๐ฅ, ๐ฆ, ๐ง. We can then subtract vector ๐ from
both sides of this equation. The left-hand side becomes negative
two, four, three minus three, five, three.

Negative two minus three is equal
to negative five. Therefore, our ๐ข-component of
vector ๐ is negative five. Four minus five is equal to
negative one, so the ๐ฃ-component is negative one. Finally, three minus three is equal
to zero. Vector ๐ is, therefore, equal to
negative five, negative one, zero.

We can calculate the magnitude of
this vector by squaring each of the components, finding their sum, and then square
rooting. The magnitude of vector ๐ is equal
to the square root of negative five squared plus negative one squared plus zero
squared. Negative five squared is 25,
negative one squared is one, and zero squared is equal to zero. The magnitude of vector ๐ is,
therefore, equal to the square root of 26.

In our next question, weโll find
the magnitude of a vector joining the endpoints of two other vectors.

Given that vector ๐๐ is equal to
negative five ๐ข plus two ๐ฃ minus four ๐ค and vector ๐๐ is equal to four ๐ข plus
four ๐ฃ plus six ๐ค, determine the magnitude of vector ๐๐.

In this type of question, it is
worth drawing a diagram first. This will hopefully ensure that our
direction and signs are correct. We are given three points ๐ด, ๐ต,
and ๐ถ. We can join these to form a
triangle. In the question, we are given the
value of vector ๐๐. We are also given the value of
vector ๐๐. Our aim is to calculate the
magnitude of vector ๐๐. Therefore, our first step is to
work out vector ๐๐.

We can see that one way to get from
point ๐ด to point ๐ถ is via point ๐ต. Therefore, vector ๐๐ is equal to
vector ๐๐ plus vector ๐๐. Vector ๐๐ is, therefore, equal to
negative five ๐ข plus two ๐ฃ minus four ๐ค plus four ๐ข plus four ๐ฃ plus six
๐ค.

We can add two vectors by adding
the individual components separately. Negative five ๐ข plus four ๐ข is
equal to negative ๐ข. Two ๐ฃ plus four ๐ฃ is equal to six
๐ฃ. Finally, negative four ๐ค plus six
๐ค is equal to two ๐ค. Vector ๐๐ is equal to negative ๐ข
plus six ๐ฃ plus two ๐ค.

The magnitude of any vector can be
found by squaring the ๐ข-, ๐ฃ-, and ๐ค-components, finding their sum, and then
square rooting the answer. This means that the magnitude of
vector ๐๐ is the square root of negative one squared plus six squared plus two
squared. Negative one squared is equal to
one, six squared is 36, and two squared is equal to four. One, 36, and four sum to 41. Therefore, the magnitude of vector
๐๐ is the square root of 41.

In our final question, we will find
the magnitude of the difference of two vectors.

If vector ๐ is equal to four ๐ข
plus four ๐ฃ minus five ๐ค and vector ๐ is equal to three ๐ข minus ๐ค, determine
the magnitude of vector ๐ minus vector ๐.

Our first step in this question is
to calculate vector ๐ minus vector ๐. This is important as a common
mistake here would be to think that the magnitude of vector ๐ minus vector ๐ is
equal to the magnitude of vector ๐ minus the magnitude of vector ๐. This, however, is not true.

To work out vector ๐ minus vector
๐, we simply subtract the individual components separately. Four ๐ข minus three ๐ข is equal to
one ๐ข, or just ๐ข. There is no ๐ฃ-component in vector
๐. Therefore, we are left with four
๐ฃ. Negative five ๐ค minus negative ๐ค
is equal to negative four ๐ค. This is because negative five minus
negative one is the same as negative five plus one, which is equal to negative
four.

We now need to calculate the
magnitude of this vector. We know that for any vector written
in the form ๐ฅ๐ข plus ๐ฆ๐ฃ plus ๐ง๐ค, its magnitude is equal to the square root of
๐ฅ squared plus ๐ฆ squared plus ๐ง squared. This means that the magnitude of
vector ๐ minus vector ๐ is the square root of one squared plus four squared plus
negative four squared.

Both four squared and negative four
squared are equal to 16, so we are left with the square root of one plus 16 plus
16. This is equal to the square root of
33. The magnitude of vector ๐ minus
vector ๐ is the square root of 33.

We will now summarize the key
points from this video. For any vector written in the form
๐ฅ๐ข plus ๐ฆ๐ฃ plus ๐ง๐ค, the magnitude of the vector is the square root of ๐ฅ
squared plus ๐ฆ squared plus ๐ง squared. We square the individual components
of ๐ข, ๐ฃ, and ๐ค, find their sum, and then square root the answer. The answer for the magnitude will
always be positive.

When adding or subtracting two
vectors, we add or subtract the individual components separately. We also saw that the magnitude of
the sum or difference of two vectors is not the same as the magnitude of their
individual parts. The magnitude of vector ๐ plus
vector ๐ is not equal to the magnitude of vector ๐ plus the magnitude of vector
๐.