Question Video: The Magnitude of a Horizontal Vector Mathematics

Video Transcript

Find the magnitude of the vector 𝐯 shown on the grid of unit squares below.

In this question, we’re given a vector 𝐯 and we’re asked to find the magnitude of this vector. And to do this, we’re given a graphical representation of our vector 𝐯 on a grid of unit squares. So, to do this, we’re first going to need to recall what we mean by the magnitude of a vector. And we recall when represented graphically, the magnitude of a vector is the length of this vector. And because we’re told that this is a grid of unit squares, the length of the sides of all of these squares is going to be equal to one.

So because our vector goes through four of these unit squares, its length is going to be four. In other words, the magnitude of this vector is four. So, we could stop here answering the question as the magnitude of 𝐯 is equal to four. However, this is not the only way we could answer this question. We could also find the component definition of 𝐯 and then find the magnitude this way.

Recall we can also represent a vector 𝐮 component-wise. We say that 𝐮 is the vector 𝐮 sub 𝑥, 𝐮 sub 𝑦 if 𝐮 sub 𝑥 is the horizontal change in our vector and 𝐮 sub 𝑦 is the vertical change in our vector. And to find both of these values from the graphical representation of our vector, we need to recall that our vector starts at the tail of our arrow. This is called the initial point of our vector. And our vector ends at the head of the arrow. This is also called the terminal point of our vector.

And as we travel from the initial point of our vector to the terminal point of our vector, we can see that we move four units to the left. And if we think of a number line or a pair of axes, when we move to the left we’re decreasing our values. So, we’ve decreased the horizontal value by four. So, the horizontal component of our vector 𝐯 is going to be negative four because we’ve decreased the horizontal value by four. We can also see on our diagram when we move from the initial point of our vector to the terminal point of our vector, we don’t move vertically at all. The vertical change is zero. So, the vertical component of our vector 𝐯 is zero.

We can then use this to find the magnitude of our vector. Recall the magnitude of a vector is the sum of the squares of the components of this vector. So, the magnitude of the vector 𝑎, 𝑏 is the square root of 𝑎 squared plus 𝑏 squared. Applying this to our vector 𝐯, we get the magnitude of 𝐯 is the square root of negative four all squared plus zero squared. And we can calculate this. We get the square root of 16, which is equal to four.

Therefore, we were able to show two different ways of finding the magnitude of a vector 𝐯 on a grid of unit squares shown graphically. In both cases, we were able to show the magnitude of this vector 𝐯 was four.