Project 1
Getting to know vectors
Sections 12.1-12.4
DUE: Wednesday January 25 at the beginning of class
Late homework will not be accepted
Objectives: understand fundamental properties of vector quantities, begin to study 3-D
geometry using vectors, and apply the dot product and the cross product.
Directions: Answer all questions on lined paper. Use only pencil and write on one side
of the paper only. Include your name, section number, and recitation time at the top of
EACH page. When explanations are necessary, write in complete sentences and use
proper English and mathematical vocabulary to express your thoughts. Disorganized
sloppy, work will not even be considered so take this as an opportunity to do your best
work. Ask yourself, would you want a potential employer to look at your final product?
12.1—12.2 Three Dimensional Geometry and Vectors
Short Answer (3 points each):
1.) Describe the 3-D region given by y=z and include a sketch.
2.) Compute the equation of a sphere with center (1,2,3) and radius 4. What is the
distance from the center of the sphere to the origin? Does sphere cross the x-axis? If so, at
which points? (three points each part)
3.) What is the difference between a scalar and a vector? Give an example of each.
4.) How can we tell if two vectors are parallel?
Looking Ahead: Working with Surface in Three Dimensions (1 point
each)
Try to draw the 3-D surfaces described by the following equations. It might be easier to
think about the surfaces first before attempting to draw anything. It also helps to think
about what ‘slices’ in the z-plane look like, that is, set z equal to a constant and consider
the resulting curve in the x-y plane…then imagine these all along the z-axis to visualize
the entire surface.
1.) x 2  y 2  9
2.) x 2  y 2  z 2
Quick Computations with Vectors (2 points each):
Suppose P = (0,2,2), Q=(3,2,4), R=(2,1,0).
1.) If the vector a has initial point P and terminal point Q, find the position vector of a.
2.) If the vector b has initial point P and terminal point R, find the position vector of b.
3.) Express a and b in terms of the standard basis vectors i,j,k.
4.) Compute the following: |a|, |b|, 2a-b, a+j (2 points for each part)
12.3 The Dot Product
Short Answer (3 points):
1.) Explain why a  b  c does not make sense.
2.) How can the dot product be used to find the angle between two vectors? How can this
be helpful to determine if two vectors are perpendicular?
3.) What is the expression for the vector projection of b onto a? Demonstrate the concept
visually in 2-D.
Quick Computations with the Dot Product (3 points each):
Consider the vectors a=<1,2,4> and b = <-2,3,6>
1.) Determine, using the dot product, if the vectors are orthogonal, parallel, or neither.
2.) Find the angle between the two vectors.
3.) Find the scalar and vector projection of a onto b and of b onto a.
12.4 The Cross Product
Short Answer(3 points each):
1.) How is the vector c=a X b related to a and b? If a X b = 0, what can we say about a
and b?
2.) How do you find the area of a parallelogram determined by a and b?
3.) How do you find the volume of a parallelepiped determined by a,b, and c?
4.) How do you find the cross product between two vectors if you know their lengths and
the angle between them? What if you know their components?
Quick Computations with the Cross Product (5 points each):
1.) Find the cross product of a = i-j+k and b = -i+2j+3k and verify that it is orthogonal to
a and b.
2.) Find a unit vector orthogonal to both <1,1,1> and <0,-3,6>.
LAST PART COMING SOON….