MAT 267 Test 1 – Review
10.1 The three dimensional coordinate system, distance formula, and equations of
spheres. Be able to sketch regions corresponding to given equations and inequalities.
10.2 Definition of vectors (know both notations), length of vectors, operations with vectors.
Be able to find unit vectors and know the special unit vectors. Practice on application problems
with force and velocity.
10.3 The dot product. Know how to find the angle between two vectors, know how to decide
whether two vectors are orthogonal. Know how to find scalar and vector and orthogonal
projections of a vector in the direction of another vector.
10.4 The cross product. Know the determinant form definition, and remember that the cross
product of a and b is a vector orthogonal to both a and b in the direction of the right hand rule.
Know how to decide if two vectors are parallel. Be able to use cross product to calculate areas of
triangles and parallelograms. Understand how the scalar triple product gives the signed volume
of a parallelepiped.
10.5 Know the vector and parametric form of the equation of a line. Be able to decide
whether two lines are parallel, orthogonal, or skew. Know the vector and scalar form of the
equation of a plane. Be able to find:
* the angle between two planes (it is the smaller angle between the normal vectors)
* the line of intersection of two planes (the direction of the line is the cross product of the normal
vectors, a point on the line is any point that belongs to both planes)
* the point of intersection of a plane and a line (substitute the parametric equations of the line in
the equation of the plane and solve for the parameter)
* the distance between two parallel planes (find a point on one of the planes and use the distance
formula from a point to a plane; if you don't remember the distance formula from a point to a
plane you can use some right triangle trigonometry together with vector and scalar projections to
find the answer)
* the distance between a line and a parallel plane (find a point on the line and use the distance
formula from a point to a plane)
10.6 Be able to identify cylinders and quadric surfaces in standard form. Know how to find the
traces in the x = k, y = k and z = k planes. Practice sketching these surfaces in three dimensions.
Be able to match graphs to equations.
10.7 Know what a vector function is, how to find limits of vector functions and how to
determine whether they are continuous or not. Practice sketching in both 2 and 3 dimensions.
Know how to identify parametric curves, be able to match their graphs and equations. Know how
to find parametric equations for a space curve given as the intersection of two surfaces. Know
how to find the derivative of a vector function and the differentiation rules. Be able to use the
derivative for finding the equation of the tangent line to a space curve. Be able to find the unit
tangent vector to a curve. Know how to find the integral of a vector function.
10.8 Know how to find the arc length of a vector function. Be able to find T, N and B at a
given point.
10.9 Given the vector function describing the position of an object, be able to find the velocity
and the acceleration vector functions and vice versa. Understand how to split up the
acceleration vector into normal and tangential components.
NOTE: All the assigned homework problems, the quizzes and the examples worked out in class
are fair questions on the test.