The document discusses position vectors in 3D coordinate systems. It defines a position vector r as a fixed vector that locates a point P(x,y,z) in space relative to another point, such as the origin O. The components of r are the distances from O to P along the x, y, and z axes. As an example, it shows calculating the position vector and length of an elastic rubber band attached between points A and B to determine the band's direction from A toward B.

1. 2.7 Position Vectors
x,y,z Coordinates
- Right-handed coordinate system
- Positive z axis points upwards, measuring
the height of an object or the altitude of a
point
- Points are measured relative to the
origin, O.

2. 2.7 Position Vectors
x,y,z Coordinates
Eg: For Point A, xA = +4m along the x axis,
yA = -6m along the y axis and zA = -6m
along the z axis. Thus, A (4, 2, -6)
Similarly, B (0, 2, 0) and C (6, -1, 4)

3. 2.7 Position Vectors
Position Vector
- Position vector r is defined as a fixed vector
which locates a point in space relative to another
point.
Eg: If r extends from the
origin, O to point P (x, y, z)
then, in Cartesian vector
form
r = xi + yj + zk

4. 2.7 Position Vectors
Position Vector
Note the head to tail vector addition of the
three components
Start at origin O, one travels x in the +i direction,
y in the +j direction and z in the +k direction,
arriving at point P (x, y, z)

5. 2.7 Position Vectors
Position Vector
- Position vector maybe directed from point A to
point B
- Designated by r or rAB
Vector addition gives
rA + r = rB
Solving
r = rB – rA = (xB – xA)i + (yB – yA)j + (zB –zA)k
or r = (xB – xA)i + (yB – yA)j + (zB –zA)k

6. 2.7 Position Vectors
Position Vector
- The i, j, k components of the positive vector r
may be formed by taking the coordinates of the
tail, A (xA, yA, zA) and subtract them from the
head B (xB, yB, zB)
Note the head to tail vector addition of the
three components

7. 2.7 Position Vectors
Length and direction of
cable AB can be found by
measuring A and B using
the x, y, z axes
Position vector r can be
established
Magnitude r represent
the length of cable

8. 2.7 Position Vectors
Angles, α, β and γ
represent the direction
of the cable
Unit vector, u = r/r

9. 2.7 Position Vectors
Example 2.12
An elastic rubber band is
attached to points A and B.
Determine its length and
its
direction measured from A
towards B.

10. 2.7 Position Vectors
Solution
Position vector
r = [-2m – 1m]i + [2m – 0]j + [3m – (-3m)]k
= {-3i + 2j + 6k}m
Magnitude = length of the rubber band
r= (− 3)2 + (2)2 + (6)2 = 7m
Unit vector in the director of r
u = r /r
= -3/7i + 2/7j + 6/7k

11. 2.7 Position Vectors
Solution
α = cos-1(-3/7) = 115°
β = cos-1(2/7) = 73.4°
γ = cos-1(6/7) = 31.0°