Math 231 Fall 2008 Review sheet for Exam 1 Our midterm exam will cover Chapters 13 14 Some important skills Section 13 1 Three dimensional Coordinate Systems Find the distance from a point P to the coordinate planes or the coordinate axes Find the distance between two points in R3 The general equation of a plane and general equation of a sphere in R3 Find the equation of a sphere Find the radius and the center of a sphere given by a quadratic equation Sections 13 2 13 3 Vectors The dot product Adding vectors triangle law parallelogram law scalar multiplication Find the dot product of two vectors Given a vector v in R3 find its magnitude length or norm Find a unit vector that has the same direction as a given one Apply the cosine formula Determine whether two vectors are parallel perpendicular or neither Scalar projection and vector projection of a vector b onto a Find the work done by a force acting on an object Section 13 4 The cross product Find the cross product of two given vectors Find the area of the parallelogram Find the area of of a triangle Find a vector orthogonal to a given plane Find the volume of a parallelepiped determined by vectors or points Torque about a point Section 13 5 Equations of lines and planes Find a vector parametric or symmetric equations of a given line L Decide whether three given points lie in the same line Find the equation of a given plane Decide whether two lines lines are parallel intersecting or skew 1 Decide whether four given points lie in the same plane Find the distance from a point P to a given plane Find the distance between two parallel planes Section 13 6 Cylindrical and Quadratic Surfaces Find traces of a surface S in the planes x k y k or z k Find the traces of a given quadric surface and give a rough sketch Identify cylinders in the space Identify quadric surfaces ellipsoids paraboloids cones hyperboloids Section 14 1 Vector Functions and Space Curves Describe a curve given by a vector function Find limits and derivatives of vector functions Find the parametric equations of a line segment from P to Q Find the vector function that represents the curve of intersection of two given surfaces Section 14 2 Derivatives and Integrals of Vector Functions Find the derivative of a given vector function Find parametric or symmetric equation of the tangent line to a curve C at a given point Find the unit tangent vector of a vector function r at a given point P Section 14 3 Arc Length and curvature Find the length of a space curve Parametrize a curve with respect to arc length Find the curvature of a smooth space curve r hf t g t h t i Find the normal vector N and binormal vector B at a given point of a space curve defined by r t Section 14 4 Velocity and acceleration Find the velocity speed and acceleration of a moving particle P t Find the normal and tangential components of acceleration Given the acceleration initial velocity and position of a particle find its velocity and its position vector at time t 2