Math 241 F1H: Exam 1 Syllabus and Checklist

Checklist of topics, concepts, and formulas

12.1/12.2 Vectors

• vectors in 2, 3 and n dimensions; magnitude (length), direction, coordinates, vectors versus scalars, right-hand/screw-driver rule
• position vector of a point
• addition of vectors (geometric and algebraic interpretation)
• multiplication of vectors by scalar (geometric and algebraic interpretation)
• unit vector; finding a unit vector in a given direction (normalizing a given vector)
• standard basis unit vectors

12.3: Dot products

• algebraic and geometric definitions
• angle formula
• properties of dot product
• scalar projection of one vector onto another (comp_ab)
• vector projection of one vector onto another (proj_ab)

12.4: Cross products

• algebraic and geometric definitions, computation of 3x3 determinants, screw-driver/right-hand rule
• geometric properties of cross product
• algebraic properties of cross product
• scalar triple product
• area of a parallelogram or triangle
• volume of a parallelepiped or pyramid

12.4: Lines and planes in space

• vector parametric, symmetric equations of a line
• vector, scalar/linear equations of a plane
• normal vector of a plane

Additional topics: n-dimensional space

• The spaces Rⁿ
• Basic arithmetic with vectors in Rⁿ: norm, addition, subtraction, scaling
• Dot product in Rⁿ. Definition and properties
• The Cauchy-Schwarz Inequality (vector and scalar form)
• The Triangle Inequality (vector and scalar form)

13.1/13.2: Vector-valued functions and curves in space

• vector functions and space curves
• derivatives of vector functions: algebraic definition (componentwise derivative), geometric interpretation, limit definition
• rules for derivatives
• derivatives of vector functions of constant magnitude
• integrals of vector functions
• motion is space: position, velocity, speed, acceleration

13.3: Curvature and acceleration

• arc length of a space curve
• T, N, B vectors, definitions and properties
• curvature (two formulas)
• normal and osculating planes

13.4: Motion in space

• position, velocity, speed, acceleration
• tangential and normal components of acceleration (two formulas each)

14.1: Functions of multiple variables

• Functions of several variables
• Domain
• Graph
• Level curves and level surfaces

14.3: Partial derivatives

• Definition of partial derivatives, and notations
• Interpretations: slope, rate of change
• Higher order partial derivatives
• Clairot's Theorem

Material that will NOT be on the exam

• Section 12.6
• Formal definition of limits and continuity of vector functions and functions of several variables (part of 13.1 and 14.2)
• Direction angles (p. 781 bottom - p. 782 middle)
• Formula for triple cross product a x (b x c) (Property 6 in the box on p. 790)
• Formula (11), p. 834, for curvature of a plane curve
• Formula (4), p. 841, for motion of projectiles
• Kepler's Laws (p. 845 - 846)

Typical computational tasks

1. Check if two vectors are (a) parallel, (b) perpendicular. (12.2)
2. Compute the angle between two vectors. (12.2)
3. Given a vector, find a unit vector with the same direction. (12.1)
4. Compute the dot product and the cross product of two vectors (12.2, 12.3)
5. Given two vectors, find the scalar and vector projections of one along the other (12.2)
6. Find a vector that is orthogonal to two given vectors. (12.3)
7. Compute the area of a parallelogram/triangle determined by two vectors (or 3 points). (12.3)
8. Compute the volume of a parallelepiped/pyramid determined by three vectors (or 4 points). (12.3)
9. Determine whether three vectors are in the same plane ("coplanar") (12.3)
10. Given a point and a direction vector, or two points, find the equation of the line going through these points. (12.4)
11. Find the equation of a plane determined by (a) a point and a normal vector, (b) three points, (c) a point and two vectors in the plane. (12.4)
12. Given symmetric or parametric equations of a line, find a direction vector for the line. (12.4)
13. Given the linear equation of a plane, find a normal vector to that plane. (12.4)
14. Given two lines, determine whether they are skew, parallel, or intersect. If they intersect, determine the point of intersection. (12.4)
15. Given a line and a plane, determine whether they intersect in a single point (and find the intersection point in this case), or are parallel. (12.4)
16. Given two planes determine whether they are parallel or intersect. In the latter case, find the line of intersection and the angle between the planes. (12.4)
17. Find the distance between (a) a point and a plane, (b) two skew lines, (c) a point and a line. (Solve such distance problems by geometric reasoning (using projections as key tool), rather than memorized formulas.)
18. Find the derivative and the integral of a vector function by componentwise derivation. (13.2)
19. Find the derivative of a vector function using the rules for derivatives (product rule for dot/cross products, etc.) and properties of dot/cross products (13.2)
20. Given the position function r(t) of a particle, find its velocity, speed, acceleration. (13.4)
21. Given the force acting on a particle or its acceleration, and its initial position and initial velocity, find its position function r(t). (13.4)
22. Find the arc length of a space curve. (13.3)
23. Find the curvature of a given curve. (13.3) (know both formulas for K))
24. Given a curve (described by a vector function r(t)) find the vectors T, N, B, and the curvature at a given point on the curve. (13.3)
25. Given the velocity and acceleration of a particle, find the tangential and normal components of its acceleration. (13.4)
26. Find the tangential and normal components of the acceleration if curvature and speed are given. (13.4)
27. Finding the graph of a given function (14.1)
28. Computing level curves and level surfaces to a given function (14.1)
29. Compute partial derivatives of first and second order. (14.3)

Relevant review questions from the text

• Chapter 12, p. 812-813: Concept Check: All questions except 18, 19.
  True/False Quiz: All questions except 13.
• Chapter 13, p. 849-850: Concept check: All questions except 6(d), and 9 (Kepler's Law).
  True/False Quiz: All questions except 7, 11.