The vector 𝐯 is shown on the grid of unit squares below. Find the value of the magnitude of 𝐯.
In the process of reading out the problem, I’ve given away the fact that this piece of notation here the vector 𝐯 in between the two vertical lines represents the magnitude of 𝐯. One way of thinking of vector quantities is as quantities which have both magnitude and direction.
For example, if I tell you that a certain town is three miles northeast of where I am, that’s a vector information because I’m telling you not only how far away it is, but in what direction. You might know that three miles northeast is a displacement. The magnitude of that displacement is just the distance, three miles. So we throw away that direction information.
Have a look at the vector in the diagram, this is an abstract geometric vector, which could represent a displacement or a velocity or a force or something else. For a geometric vector like the one we have, the magnitude is just the length. How do we find the magnitude or length of this vector? We use the grid of unit squares that it is drawn on.
We draw a right triangle, the base of which lies on top of four unit squares and so has a length of four units. And in a similar way, we see that this side has a length of three units. The magnitude of 𝐯 is then the length of the hypotenuse, which we can find using the Pythagorean theorem.
Applying the Pythagorean theorem, we get that the length of the vector is the square root of three squared plus four squared. Three squared plus four squared is 25. And so the length is the square root of 25, which is five. Remember this is not only the length of the hypotenuse in the diagram. This is the magnitude of the vector that we wanted to find.
To specify our vector exactly, we need not only this magnitude five, which tells us how long the vector is, but also the direction in which the vector is pointing. Vectors have both a magnitude and a direction. And in this problem, we’ve seen how to calculate the magnitude when the vector is represented geometrically in the plane.