(5) Finding the magnitude or length of a vector

This drawing shows you how we set about doing this.
Notice the right-hand coordinate system but with the z-axis vertical this time.)
how to find the length of a vector
The formula for the length OP or |p| is a kind of 3 dimensional form of Pythagoras' theorem.
Here's how the working would go if we had a = 4 and b = 3 and c = 12.
Using Pythagoras' theorem in the yellow right-angled triangle we get
OQ = the square root of (16 + 9) = 5

and then, using Pythagoras' theorem in the vertical green right-angled triangle, we get
OP = the square root of (25 + 144) = 13.

Alternatively, in just one step of working, we have

OP = the square root of (16 + 9 + 144) = 13.

A reminder on unit vectors
Unit vectors are vectors which have unit length. They are important in many applications. In three dimensions, the vectors i, j and k which run along the x, y and z axes respectively, all have unit length.
Sometimes, for example when working with planes, it is necessary to find a unit vector in the same direction as a given vector.
Suppose you need the unit vector in the same direction as  u = 2i - j + 2k  or   u = (2, -1, 2) written as a row vector.
The length or magnitude of u is given by the square root of  (4 + 1 + 4) = 3.
Also, the required vector must be parallel to u.
So we can get the vector we want by just scaling down u by a factor of 3.
It is  1/3(2i - j + 2k) = 1/3(2, -1, 2) = (2/3, -1/3, 2/3).
Unit vectors are often written with a little hat on top, so here we would have found û.
Here's a practical application which uses components and magnitude.
If a body is acted on by three forces P, Q and S whose lines of action all pass through the same point, find the resultant force R, and its magnitude |R| if
P = 3i - 5j + 2k
Q = - 2i + 4j + 4k
S = 3i - 3j - 4k.

P + Q + S = R = (3 - 2 + 3)i + (- 5 + 4 - 3)j + ( 2 + 4 - 4)k = 4i - 4j + 2k
and |R| = the square root of (16 + 16 + 4) = 6.

We might also need to know what angles this resultant makes with the x, y and z axes. Finding these angles will come as a spin-off from discovering how we can multiply two vectors together.
But how can we actually do this since the two vectors can have different directions?


Next

or back to the previous section

or return to the vector homepage.