(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
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
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