Wize University Physics Textbook (Master) > Force Vectors (Advanced Info)
Position Vectors (r)
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Position Vector (r) = fixed vector that locates a point in space relative to another point. Some call it the "distance" vector, as its magnitude is the distance between the endpoints.
- can be used to form a unit vector
- can be used to determine the length or distance between 2 points
IF USE THE ORIGIN AS THE STARTING POINT:
ex: so two points are O (0,0,0) and A (Ax, Ay, Az)
IF the ORIGIN is NOT the STARTING POINT:
ex: so two points are A (Ax, Ay, Az) and B (Bx, By, Bz), then rAB can be written as:
Wize Tip
Reduce errors by re-writing the coordinates A(x, y, z) and B(x, y, z) BEFORE building the position vector. Don't pull the numbers right from the figure as you build it - easy to make mistakes!
Given coordinates A (3, 5, 6)m and B (5, -2, 1)m, find:
a) position vector r between (from) A & B
b) distance btwn A & B (which is the length of position vector r)
c) unit vector of the position vector r
d) direction cosines angles
---------------------------------------------------------------------------------
SOLUTION:
a) rAB = (5-3)i + (-2-5)j + (1-6)k
= 2i - 7j -5k m
b) distance btwn A and B will equal the length (or magnitude of the vector rAB)
rAB mag = sqrt(22 + (-7)2 + (-5)2) = 8.8 m
c) uAB = ( vector rAB / mag rAB ) = (2i-7j-5k)/(8.8)
= 0.227i=0.795j-0.568k
d) direction cosines = cos-1(component / vector mag)
alpha = cos-1(2/8.8) = 76.9°
beta = cos-1(-7/8.8) = 142°
gamma = cos-1(-5/8.8) = 124°
(ANS: (2i-7j-5k)m or <2,-7,-5>m)
(ANS: 8.8m)
(ANS: (0.227i-0.795j-0.568k))
(ANS: 76.9°, 142°, 124°)
Mark Yourself Question
- Grab a piece of paper and try this problem yourself.
- When you're done, check the "I have answered this question" box below.
- View the solution and report whether you got it right or wrong.
Determine the position vectors and distances of AB, AC, and AD.
Mark Yourself Question
- Grab a piece of paper and try this problem yourself.
- When you're done, check the "I have answered this question" box below.
- View the solution and report whether you got it right or wrong.
Given coordinates A (1,1,1,) and B(4,6,9)m, find:
a) position vector r between (from) A & B
b) distance between A & B (which is the length of position vector r)
c) unit vector of the position vector r
d) direction cosines angles