Geometry
Name_______________________
Review of Vectors: Applications & Proofs
Directions: Please show your work for each of the following problems. Take care in setting up each
scenario, given the following information. Use appropriate units.
1) Given the following coordinates for A (1, 4) and B (7,−2), find the coordinates for P using
vectors:
a. P is 1/2 the way from A to B
b. P is 1/3 the way from B to A
c. P is 4/5 the way from A to B
d. P is 1/4 the way from B to A
2) In the diagram, M and N are the midpoints of PQ and PR .
a. Express MN in terms of a and b .




b. Express QR in terms of a and b .
P

a



a

b

N
M

b


Q



3) Complete the following equations given that MN is the median of trapezoid ABCD.
a. AM  MN  ND 





4)

b. BM  MN  NC 
c.
AM  BM 
d. ND NC 
e. Add equations from a and b, and, by using c and d, simplify the results to
2MN  AD BC . What theorem about the median of a trapezoid does this equation
suggest?
R
Let A  x A , y A , B  x B , y B  and M be the midpoint of AB . Use vectors to prove the
“midpoint formula.”



5) The diagonals of quadrilateral PQRS have the same midpoint N. Use problem 4 to prove
that SP  RQ and SR  PQ . What quadrilateral theorem does this prove?
6) Point A divides MT in the ratio 2:3, that is
of v :
a. TA



b. MT

c.

MA 2
 . If v  MA , express each vector in terms
AT 3

M
A
T
AM


B
7) In the diagram, ABCD is a parallelogram. If AB  x and
AD  y , express each vector in terms of x and y .
a. BC
b. CD


 A



b. AC
d. AO
x

e. BO

O
y


C
D
8) A swimmer leaves point A swimming south across a river at 2 km/h. The river is 4 km wide
and flows east at the rate of 1 km/h. Make a vector diagram showing her resultant velocity.
Calculate her resultant speed. How long will it take her to swim across the river? How far
east does she swim?
9) Find the values of x and y given that v = xi + 2yj – 8k is perpendicular to both
a = 2i – j + k and b = 3i + 2j – 4k (hint: what must be true about the angle between the
vectors?)
10) Your friends want to borrow your jet ski for the next long weekend to use at Lake
Washington. They will be taking off from a shared dock that is 4 yards east and 2 yards
north of the shore where you parked. They head out at a velocity of 2i + 3j yards/min.
a) What is their position from your house after they’ve been gone for 5 minutes?
b) How far have they traveled in 5 minutes?
c) What is their speed?
11) A commercial jet airplane cruises at 450 mph in still air at a heading of 30˚. If there is a 40
mph wind blowing from a heading of 290˚, draw a vector model of this situation and find the
resultant heading and speed of the plane.
12) How can the plane from problem 11 adjust its heading and initial velocity so that its
resultant course and velocity will be due North at 500 mph. Draw using a vector diagram
and find the plane’s initial heading and starting velocity.