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HARVARD MATH 21A - Study Guide

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Math 21a Study Guide for Math 21a Hourly 1What follows are a list of topics and issues that may arise on the first hourlyexam. Though not exclusive, the list is provided to help you identify some of the salienttopics that the course covers in Chapter 1 and Appendix A of the text book.In any event, remember that the first houly exam is on Wednesday, October 10,from 7-9pm in Science Center lecture halls C and D (go to either one). Please arrive afew minutes before 7 as we will start timing the exam at 7.• Given a pair of vectors in R2 or R3, be able to add them, subtract them (bothanalytically and graphically), multiply them by real numbers, take their dot productand take their cross product.• Understand the basic formula for dot and cross product in terms of vector length andangle between vectors. For cross product, determine the direction of the crossproduct.• Manipulate and understand simple algebraic, geometric and trigonometric facts andformulas that involve the dot and cross product.• Identify orthogonal and parallel vectors. Find vectors orthogonal to a given vector orto a given pair of vectors.• Find normal vectors to planes, vectors tangent to planes, vectors tangent to lines andvectors normal to lines.• Given sufficient data, provide parametric and non-parametric descriptions of linesand planes.• Determine when two planes or a line and a plane intersect or a pair of lines intersector are parallel. Be able to write down an equation for the points in these intersections.• Calculate the distance from a point to a given point, line or plane.• Know how to determine the projection of one vector along another.• Write one or more parametrizations for given simple paths in R2 or R3.• Provide the velocity vector and acceleration vector for a particle with given timedependent position vector.• Provide the unit tangent vector to a given parametrized path at a given point in time.• Solve simple problems involving particles moving along parametrized paths. Forexample, be able to tell when a particle crosses a given plane, is moving tangent to agiven line, is closest to a given point.• Differentiate and otherwise manipulate dot and cross products of vector functions oftime.• Develop an expression for the length of a given parametrized path as an integral.• Sketch a graph of a given parametrized path (even in polar coordinates).• Understand the relation between polar coordinates and standard Cartesian coordinatesfor


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HARVARD MATH 21A - Study Guide

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