Math32a Midterm1 Summary The material for the midterm includes Chapter 12 except Section 12 6 Chapter 13 except Section 13 4 Sections 10 1 10 2 Bring your ID card to the exam No calculators no books no notes no cheatsheets no cell phones no computers will be allowed on the exam Below is the summary list of the concepts methods theorems statements and formulas that students should know and understand Know the basic concepts of the 2 dim and 3 dim coordinate system coordinate axes coordinate planes 4 quadrants in 2 d 8 octants in 3 d orthogonal projections onto coordinate planes and onto coordinate axes Be able to distinguish between the right handed and left handed orientations of coordinate systems similarly between the left handed and right handed triples of vectors u v w Be able to find distance between two points in R2 and R3 Be able to find the midpoint between two points in R2 and R3 Be able to draw graphs of basic equations inequalities in R2 and R3 in particular lines circles spheres cylindrical surfaces Know and understand the concepts of a vector components of a vector different representations of a vector initial and terminal point of a representation of a vector Be able to compute the components of a vector with initial point P1 and terminal point P2 Know how to compute the length magnitude of a vector Know how to add subtract vectors geometrically triangle law and parallelogram law and algebraically when the components are known Same for multiplication of vectors by a scalar Know that c u c u for any scalar c Know what a unit vector is and be able to find unit vector in a given direction 1 Know basic properties of addition and scalar multiplication p 819 Know the definition of the dot product of two vectors and it s basic properties p 825 Know that angle between two vectors can be computed by cos u v u v Know that two vectors are perpendicular orthogonal if and only if their dot product is zero Know what the directional angles of a nonzero vector are and be able to compute them Be able to compute projection of one vector onto another vector both scalar projections and vector projections Know how to compute 2 2 and 3 3 determinants Know the definition of the cross product of two vectors and it s basic properties p 836 no need to remember no 5 and 6 Know that the cross product u v is perpendicular to both u and v has length u v sin and that u v u v forms a right handed triple Be able to compute the area of a parallelogram and of a triangle with given vertices Know that two vectors are parallel if and only if their cross product is the zero vector Know the definition of the scalar triple product and vector triple product Know the formula for computing the scalar triple product and its geometric interpretation in terms of parallelepiped Be able to write vector parametric and symmetric equations of a line through a given point parallel to a given vector Be able to write vector parametric and symmetric equations of a line through two given points Be able to obtain vector parametric and symmetric equations of a line by knowing one of them Be able to write vector and parametric equations of a line segment Be able to write scalar and vector equations of a plane through a given point with a given normal vector Be able to write equation of a plane through given three points Know what it means for three vectors to be coplanar and be able to check if they are Given an equation of a line in any of the form be able to find a vector parallel to it Given an equation of a plane be able to find a vector perpendicular to it 2 Know how to find the angle between two lines and two planes Be able to find the distance from a point to a line or a plane Be able to solve other geometrical problems such as finding intersection of two planes finding plane through two intersection lines and others Be able to find derivatives dy d2 y dx dx2 of a parametric 2 dim curve Be able to find tangent lines arc length area below the curve and area of surface of revolution of 2 dim curve Know what a vector function its domain and its range are Know the correspondence between a vector function and a parametric curve Be able to parametrize common curves and intersection of two surfaces Know how to take limit derivative and integral of a vector function Know the Fundamental Theorem of Calculus for vector functions Know that r 0 t is a tangent vector to the curve Be able to write the equation of a tangent line to a given curve in 2 dim and in 3 dim t and the Know how to find the unit tangent vector T t the principle normal vector N binormal vector B t of a curve at a given point Know what the normal plane and osculating plane are and how to find their equations Know that if r t describes the position of a particle at time t then its velocity is r 0 t its speed is r 0 t its acceleration is r 00 t Know the differentiation rules for vector functions p 874 Know how to find the arc length of a parametric curve in R3 Know what the arc length function is and be able to reparametrize a curve with respect to the arc length Be able to compute curvature of a curve at a given point using any of the formulae the formulae will be given if needed Know that curvature is the reciprocal of the radius of a circle that best approximates the curve 1 in particular it s 0 for a line and 1 R for a circle of radius R 3