Unit Vectors
Just as a meter is a “unit” of length and a pound is a “unit” of weight we introduce the “unit”
vector (although not a standard quantity per se). Given a three-dimensional coordinate system:
We let i represent the X direction
( -i is the –X direction);
let j represent the Y direction
( -j is the –Y direction);
let k represent the Z direction
( -k is the –Z direction)
As shown here:
Thus, 2i is a vector 2 units long
(maybe meters, or pounds – whatever
the quantity) in the X direction.:
And -3j is a vector 3 units long in the
negative Y direction:
4i + 3j
Finally, the three dimensional vector -8i +
5j + 5k is a vector 3 units long in the +X
direction, 5 units long in the +Y direction and
2 units in the negative Z direction, like so:
The vector 4i + 3j is a vector 4
units in the +X direction and 3 units in
the +Y direction (note that it lies in the
X-Y plane). Physically, the vector may
represent a displacement of 4 meters in
the X direction and 3 meters in the Y
direction; or perhaps 4 lbs of force in
the X direction and 3 lbs of force in the
Y direction.
1. Rules of addition/subtraction: Only LIKE unit vectors combine. i's only add (or subtract)
with other i's. j’s only add/subtract with other j’s, and k’s only add/subtract with other k’s.
Example: Add
Answer:
6.2i + 3j
to
2.1i – 5.6j
8.3i – 2.6j
2. What’s the length of this vector?
Well, the “i” value is the base of the triangle and the “j” value is the height:
8.3 units
-2.6 units
Quadrant 2 Quadrant 1
- X, +Y
+X, +Y
- X, -Y
+X, -Y
Find the hypotenuse
So we just use the Pythagorean theorem to find the length:
Length  (i value) 2  ( j value) 2
Length  (8.3) 2  (2.6) 2
Length  8.7 units
(Units could be meters, feet, m/s, lbs, Newtons, etc)
3. What’s the angle of the vector?
 j value 
 i value 
  2 .6 
  tan 1 
 8.3 
  - 17.4 degrees
  tan 1 
Quadrant 3 Quadrant 4
Note that the sign of the
coefficients tell you what
quadrant of the XY plane the
vector occupies.
In our example, +8.3 and -2.6
would indicate the 4th
quadrant.
4. How do we generate these ‘vectors’?
Given a vector A = 185 lbs at 38 degrees
Convert to an i, j vector…
185 lbs
j coefficient (height)
38o
i coefficient (base)
A = (185cos38)i + (185sin38)j
A = 146i + 114j
Given a vector
B = 47 lbs at 150 degrees
47 lbs
j coefficient (height)
150o
30o
i coefficient (base)
B = (47cos150)i + (47sin150)j
B = -40.7i + 23.5j
Now if we wanted to add A + B:
146i + 114j
-40.7i + 23.5j
105.3i + 137.5j
or = -(47cos30)i + (47sin30)j
[notice you need to add the minus sign manually
for this approach]
Or we could subtract A – B:
146i + 114j
- (-40.7i + 23.5j)
186.7i + 90.5j