Midterm 1
Some basic skills you should have:
- Determine whether two given vectors are parallel or perpendicular (or neither).
- Compute the angle between two (nonzero) vectors.
- Given a vector, find a unit vector in the same direction.
- Compute the dot product and cross product of two vectors.
- Find the component of one vector in the direction of another.
- Find the projection of one vector in the direction of another.
- Find a vector that is perpendicular to two given vectors.
- Compute the area of a parallelogram determined by two vectors or by
three points.
- Compute the volume of a parallelepiped deermined by three vectors or
by four points.
- Determine whether three vectors lie in the same plane.
- Given two points in space, find the equation(s) for the line between them (i.e. parametric equations, equivalently a vector-valued
function, and implicit equations, for example symmetric equations).
- Given a point and a direction vector, find the line through that point with
the given direction vector.
- Find the equation of a plane through a given point and with a given
normal vector.
- Find the equation of a plane through three given points.
- Given an equation for a plane, find a normal vector.
- Given two lines, determine whether they are parallel (or equal), skew,
or intersect in a single point.
- Given two planes, compute the angle between them, and determine whether they
are parallel (or equal) or intersect in a line (in this case, be able to find the equation of the line!).
- Given a vector-valued function (of one variable!), find the derivative.
- Use the product formula for dot and cross products, etc. to
compute derivatives of a dot product or cross product of two vector-valued
functions of t, a scalar-valued function times a vector-valued function, etc.
- Given the position function r(t) of a particle moving in space,
compute its velocity, speed, acceleration, and scalar acceleration.
- Given the acceleration function and initial position and velocity,
compute the position function r(t) of a particle.
- Compute the arclength of a space curve.
- Given a vector-valued function r(t), compute the unit tangent vector.
- Given a sphere described in words (for example, "the sphere
with center (0,1,-1) and radius 3") find an equation for it.
- Given an equation of a sphere, find its center and radius.
Here are some problems about those basic skills
(with solutions).
What is fair game for the midterm: Chapter 10 Sections 10.1-10.5.
Chapter 11 Sections 11.1-11.3, Section 11.4 just up through Example 4.2
(so pages 887 through the top of 889). The examples in Section 11.5 of
computing unit tangent vectors (nothing else from that section!).
You may find midterms and review information from the Math 241 I taught
last year
here.
Warning: this year's exams may or may not look anything like last year's!
Main course page.