Name:________________________________
Vectors III
3-D Vectors.
All of the vector rules that we’ve used for two dimensions also work for three (and four, five, six, etc.).
Try these problems to gain familiarity and see why 3D vectors make the problems from last unit
potentially much easier.
1) The following three vectors can be added to form vector A. What is this vector?
< 1,2,3 >
< -2, 4, 5>
<4, -10, -6 >
2) The following three vectors can be added to form vector B. What is this vector?
< 3, 2, 3 >
< -4, 1, -5>
<-1, -5, 1 >
3) The following three vectors can be added to form vector C. What is this vector?
< 2, 4, 6 >
< 2, -2, -10>
< 4, -3, -10 >
4) Determine the magnitude of vectors A, B, and C.
Find the dot product of each vector into the others. What does this tell you about the three vectors?
5) Vector D= <3, 2, -4> Find the magnitude of vector D and then find a unit vector that points in the
same direction as D.
6) A square based pyramid comes to a point directly above one of the corners of the square. The height
of the pyramid is 5 cm, and the side lengths of the base are 6 cm. Use vectors to find the angle between
the pyramid edge that connects the top to the opposite corner of the square base and one of the two
sides of the square base that meet at that point.
7) a) Take the dot product of the following two vectors N= <2,3,-4> M= <-1,3,5>
b) What is the angle between these vectors?
c) Find a vector that is perpendicular to N.
8) A certain house consists of a cube with a triangular prism on top of it (sounds confusing, just look at
the house model). All edges of the house are the same length. Find the angle between the following two
segments using vectors:
Segment one – connects one of the bottom corners of the house with the peak of the roof on the
opposite side.
Segment two – connects that same bottom corner of the house to the opposite corner across the cube.
9) Find the angle formed by the intersection of the two diagonals of a cube.
10) Find a vector that has a length of 3 and points in the opposite direction of the following vector:
<-2,4,8,-3>
Review Problems:
1) Three wolves are fighting over a moose leg (D). They are all pulling outward at the same time. The
wolf pulling in the direction of A is exerting 180 pounds of force, the one in the direction of B is exerting
150 pounds of force, and the third wolf (heading towards C) is using 190 pounds of force. What is the
resultant direction that the moose leg is moving?
B
125 deg.
A
D
90 deg.
C
2) Quentin is trying to pull a disabled snowmobile to the lodge. The snowmobile can be pulled with a
force of 120 pounds parallel to the ground. If Quentin is pulling at an angle of 25 degrees to the ground,
with how much force (in pounds) must he pull?
3) In the diagram AT:TC = 1:2. Express each of the following in the form ru  sv , where u  BA and
v  BC :
a) AB
B
C
A
T
b) AC
c) AT
d) BT