Unit 1 Vector Geometry Introduction Vectors are first introduced as geometric objects, namely as directed line segments or arrows. Operations of addition, subtraction, and multiplication by a scalar (real number) are introduced for these directed line segments. Rectangular Cartesian coordinate systems are then introduced and used to give an algebraic representation for the vectors. The operations of addition, subtraction, and multiplication by a scalar are defined algebraically. Two new operations known as a dot product and a cross product are defined. Some theorems from Euclidean geometry are proved using vector methods. The standard basis for three-dimensional space is introduced as well as the concept of a vector in a higher dimensional space. Note on notation It is important to distinguish between vectors and scalars as they are different kinds of entities. In typewritten text, vectors are denoted by bold face type such asuorvwhile scalars are denoted by italic type such asaorb. In handwritten script, this way of distinguishing between vectors and scalars must be modified. It is customary to leave scalars as regular handwritten script and to modify the symbol used for vectors by either underlining the letter representing the vector, such asu or v , or placing an arrow above the letter, such as u or v. GG Learning objectives Upon completion of this unit you should be able to: •define a vector geometrically and algebraically; •add two or more vectors geometrically and algebraically; •subtract two vectors geometrically and algebraically; •multiply a vector by a scalar; •construct a rectangular Cartesian coordinate system and plot points on it; •compute the dot product of two vectors; •use the dot product to calculate the length of a vector; •use the dot product to find the angle between two vectors; •compute the cross product of two vectors; •use the cross product to find a vector perpendicular to two given vectors; •use the cross product to find the area of a parallelogram and a triangle; •use the cross product to calculate the volume of a parallelepiped; •write any vector in three-dimensional space as a combination of standard basis vectors; and •work with vectors in anm-dimensional space where> m3. Vector Geometry and Linear Algebra MATH 1300Unit 11
