Name _____________________
Worksheet 8.3
Vector Applications
Using Vectors to find the Centroid of a Triangle
1. Draw a triangle ABC, labeling the coordinates of this triangle A  ( xA , y A ) , etc.
x 
x 
x 
2. Draw the three vectors OA   A  , OB   B  , and OC   C 
 yA 
 yB 
 yC 
3. Find the following three vectors in terms of xA , y A , xB , yB , xC , yC . The first one is done as an example.
 x   x   x  xA 
AB  OB  OA   B    A    B

 yB   y A   yB  y A 
BC 
CA 
4. Label the midpoint of each side
is the midpoint of CA.
ABC such that L is the midpoint of AB, M is the midpoint of BC and N
1
AB . This is simple scalar multiplication. Use this information to find the following
2
three vectors, again in terms of xA , y A , xB , yB , xC , yC
5. Note that AL 
AL 
BM 
CN 
6. Now find the coordinates of the midpoints by treating them as vectors. Again, the first one is done as an
example. The result should hopefully make intuitive sense.
 x A  xB 
 x  1  x  xA   2 
L  L  OL  A  AL   A    B


 y A  2  yB  y A   y A  yB 
 2 
M
N
7. Now find vectors that represent the three medians of the triangle.
 x A  xB 
 x A  xB

 2   xC   2  xC 
CL  L  C  
   

 y A  yB   yC   y A  yB  y 
C

 2 
 2
AM 
BN 
8. Now, for a moment recall that we once showed that the three medians of a triangle intersect at a single
point called the centroid (the center of mass), and that this point is two thirds from a vertex to the opposite
side’s midpoint. Label the centroid G and find the following three vectors.
CG 
AG 
BG 
9. (The payoff to all of this work). Find the location of the centroid. If you are ambitious, find it three
different ways, and thus show that all three medians do intersect at this point. Note the elegance of the
result.
Vectors and Trig
A vector in the plane has magnitude and direction. This information can be provided directly or by
representing the vector by two coordinate components. The x-coordinate is the horizontal component and
the y-coordinate is the vertical component.
 20 cos(30) 
A vector of magnitude 20 at 30 to the horizontal yields the vector 

 20sin(30) 
Draw a diagram illustrating this.
Find the vector with magnitude of 12 at 60° below horizontal (and it points to the right). Again, sketch a
picture.
If a vector is given in component form we can find its magnitude and direction.
b
tan 1
2
2
a 
a
The vector   has magnitude a  b and direction
b
 
Draw a diagram that demonstrates these ideas.
Find the component form of the following vectors
(a) Magnitude 10 direction 135
(b) Magnitude 4 direction 60
(c) Magnitude 5 direction 270
Find the magnitude and direction of the following vectors
 3
(1)  
 4
  1
( 2) 
1
 2 
(3) 
  2
A unit vector is a vector with magnitude 1
(a) What are the unit vectors in the direction of the x and y axes?
5
(b) Give a unit vector in the direction of  
12 
3
1
(c) If a    and b    , give a unit vector in the direction of b – a
 2 
4