MAT 267
TEST 1 – REVIEW
NOTE: Below, the Examples and the hw refer to the Examples in the textbook and the homework assigned from the
textbook.
10.1 The three dimensional coordinate system.
Know how to:
• Find the distance between two points (see WeBWorK #2, Example 3) and the distance from a point to any of the
coordinate axis.
• Find the equation of a sphere (WeBWorK #5, 6, 7, 8, 9,10 Examples 4,5,6 )
• Sketch simple surfaces such as planes and/or cylinders. (WeBWorK #4, Examples 1, 2)
10.2 Definition of vectors.
Know how to perform operations with vectors both geometrically (hw #1, 2, 3) and using components (WeBWorK #1,
Examples 1, 2, 5).
• Know the special unit vectors i, j and k.
• Be able to find unit vectors (WeBWorK #2, Example 6) and, more generally, vectors with a given length in the direction
of a given vector (hw 18)
• Given the coordinates of two points A and B, know how to find the components of the vector 
AB (Example 3)
• Know how to express the components of a 2D vector in terms of its length and the angle with the positive x-axis (polar
form).
• Practice on application problems with force and velocity (WeBWorK #3, 4, 5, Example 7).
•
10.3 The dot product.
• Know the definition(s) (WeBWorK 1, 2, 3, Examples 1,2 ).
• Know how to find the angle between two vectors (WeBWorK 3, 6, 8, Example 3).
• Know how to determine whether two vectors are orthogonal, parallel or neither (WeBWorK 3, 4, 5, hw #20, Example 4)
• Know how to find scalar and vector projections of a vector in the direction of another vector. (WeBWorK 7, 9, Example
5)
• Practice on application problems involving Work (WeBWork 10, 11, Examples 6 and 7).
10.4 The cross product.
• Know the definition (WeBWorK 1, 2, 5, hw #8, Examples 1, 2 )
• Know the formula for the magnitude of the cross product in terms of the magnitude of the vectors and the angle between
them 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 (WeBWorK 3, 4, hw # 10, 12, Example 3).
• Know how to decide if two vectors are parallel.
• Be able to use the cross product to calculate areas of triangles and parallelograms (WeBWork 6, 7, Example 4).
• Understand how the scalar triple product gives the signed volume of a parallelepiped (WeBWork 9) and how to use the
scalar triple product to determine whether three vectors are coplanar (Example 5) or whether four points lie on the same
plane (hw #34)
10.5 Equations of Lines and Planes
• Know the vector and parametric form of the equation of a line (WeBWorK 2, 3, 4, 5, 6, 7, Examples 1 and 2).
• Be able to determine whether two lines are parallel, orthogonal, or skew. If they intersect, be able to find the intersection
point. (WeBWorK 9, 10, Example 3)
• Know the vector and scalar form of the equation of a plane (WeBWorK 11, 12, 13, 19 Example 4).
• Know the distance formula from a point to a plane. (WeBWork 20, Example 7)
Be able to find:
• The equation of a plane containing three given points (WeBWorK 14, Example 5).
• 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) (WeBWorK 8, 21).
• 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) (WeBWorK 15).
• The distance between two parallel planes (find a point on one of the 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 right triangle trigonometry
together with vector and scalar projections to find the answer) (Example 8).
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).
• The angle between two planes (WebWorK 16, 17).
•
10.6 Cylinders and quadric surfaces.
Be able to:
• Identify cylinders and quadric surfaces in standard form by looking at their traces in the x = k, y = k and z = k planes.
Practice sketching these surfaces (WeBWorK 1, hw #9, 10, 18, 19, 20, Examples 1- 6).
• Find the standard form of the equation of a quadric surface by completing the square (hw #25, 26, Example 7).
• Match graphs to equations. (WeBWorK 2)
10.7 Vector Functions and Space curves
Know how to:
• Graph simple 2D vector functions (hw #6)
• Find the domain and limits of a vector function and how to determine whether it is continuous or not (WeBWork 1, 2, 3,
Examples 1, 2).
• Identify parametric curves and be able to match their graphs and equations. (WeBWork 4, 5, Examples 3, 4)
• Find parametric equations for a space curve given as the intersection of two surfaces. (WeBWork 6, 7, hw # 28, 29, 30,
Example 6)
• Find the derivative and the integral of a vector function (WeBWorK 8, 12, 13, Examples 8, 13).
• Find the equation of the tangent line to a space curve (WeBWorK 10, 11, Example 10).
• Find the unit tangent vector to a curve (WeBWorK 9, Example 8).
• Determine if a curve is smooth (Example 11).
10.8 Arc Length.
Know how to:
• Find the arc length of a vector function (WeBWorK 1, 2, 3, Example 1).
• Find T, N and B at a given point (WeBWorK 4, Example 6).
10.9 Motion in space: velocity and acceleration.
• Given the vector function describing the position of an object, be able to find the velocity and the acceleration vector
functions and vice versa (WeBWorK 1, 2, 3, Examples 1, 2, 3). Remember that the speed is the magnitude of the velocity.
• Practice on applied problems (WeBWorK 4-8 Examples 4 ,5, 6)
Recommended Review Problems:
Chapter 10:
Review - Concept Check (page 587): 1-12, 13 (skip symmetric equations), 14-24,26, 27(a)
Review – Exercises (pages 589-590): 1, 3, 4, 5, 6, 9, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 22, 23**, 24, 25, 27, 29, 31, 33,
37, 38, 39, 40, 41, 42, 44, 51, 52, 53.
**Note: For #23 use the lines: x=12 t , y=23 t , z=34 t and x=−16 t , y=3−t , z =−52 t .
Answers to Even recommended problems.
NOTE: This list of review problems is by no means intended to be inclusive of all the problems in the Test. Review all the
assigned homework problems, the quizzes and the examples worked out in class.