Magnitude of a vector

VECTOR ALGEBRA
a
k
ay j
j
az k
ax i
i
Figure 7.7 A Cartesian basis set. The vector a is the sum of ax i, ay j and az k.
basis vectors i, j and k may themselves be represented by (1, 0, 0), (0, 1, 0) and
(0, 0, 1) respectively.
We can consider the addition and subtraction of vectors in terms of their
components. The sum of two vectors a and b is found by simply adding their
components, i.e.
a + b = ax i + ay j + az k + bx i + by j + bz k
= (ax + bx )i + (ay + by )j + (az + bz )k,
(7.11)
and their difference by subtracting them,
a − b = ax i + ay j + az k − (bx i + by j + bz k)
= (ax − bx )i + (ay − by )j + (az − bz )k.
(7.12)
Two particles have velocities v1 = i + 3j + 6k and v2 = i − 2k, respectively. Find the
velocity u of the second particle relative to the first.
The required relative velocity is given by
u = v2 − v1 = (1 − 1)i + (0 − 3)j + (−2 − 6)k
= −3j − 8k. 7.5 Magnitude of a vector
The magnitude of the vector a is denoted by |a| or a. In terms of its components
in three-dimensional Cartesian coordinates, the magnitude of a is given by
(7.13)
a ≡ |a| = a2x + a2y + a2z .
Hence, the magnitude of a vector is a measure of its length. Such an analogy is
useful for displacement vectors but magnitude is better described, for example, by
‘strength’ for vectors such as force or by ‘speed’ for velocity vectors. For instance,
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