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### MATH 1046/2046 GLOSSARY

This page contains definitions of terms which students may not have heard of or may forget what they are. They are covered in this course as well as other math courses (such as MATH 1036/1037 or MATH 1056) or should have been covered in highschool.

The definitions included here are not necessarily complete or written in correct mathematical form. Instead they have been simplified to give a general idea of the concept while minimizing the confusion often associated with complete mathematical definitions. Therefore, the definitions should be used only to help understand a concept, not as a resource for mathematically correct definitions.

An excellent example of this is the definition of a projection. This definition is incorrect to the extent that if you were to tell it to your professor he/she would probably faint. However, the actual definition of a projection is needlessly complex, and is available in the textbook if it is required. The purpose of this glossary is to provide definitions to help you understand unclear concepts in MATH 1046/2046. So, if you are still unsure about any of the definitions you can also look in the glossary at the back of your textbooks.

# INDEX

• Cross Product
• Direction Vector
• Dot Product
• Normal Vector
• Projections
• Scalar
• (Standard) Unit Vectors
• Vector

• Cross Product
The cross product of two vectors in R3 gives a third vector which is orthogonal to the first two. Given vectors u and v, it is denoted u × v
Direction Vector
A direction vector is one that is parallel to a line or plane, and it is used in vector or parametric equations to help define a line (or two direction vectors for a plane)
Dot Product
The dot product of two vectors gives a scalar, properties of which are listed in the vector tutorial. Importantly, if the dot product equals 0, the vectors are orthogonal.
Normal Vector
A normal vector is perpendicular to a line or plane, and is used in defining the line or plane using the general or normal forms of the equation for a line or plane.
Projections
There are two types of projections: scalar projections (also called components) and vector projections. The vector projection of u on v can be thought of as the shadow that u casts on v when a light shines straight down on v. It gives a vector that lies in the direction of v and is the length of the 'shadow' cast by u. A scalar projection gives the length of the vector projection.
Scalar
A scalar is a quantity with a magnitude but no direction. They are usually denoted by lowercase letters, and in this course they usually represent real numbers.
(Standard) Unit Vectors
Unit vectors are all those vectors with a length equal to one. Standard unit vectors en are those unit vectors with a 1 as the nth component, and zeroes as all other components. So, e1 has a 1 as the first component, and 0 for all other components. In R3, e1 is parallel to the x-axis, e2 is parallel to the y-axis, and e3 is parallel to the z-axis.
Vector
A vector is a quantity with a magnitude and a direction. It is usually denoted by a bold-faced lowercase letter, or by the start and endpoint names (for example, AB) with an arrow above. A vector is geometrically interpreted as a directed line segment whose length is equal to the magnitude. Analytically, a vector is made up of two or more scalar components.
FACULTY HOMEPAGES
Alex Karassev
Ted Chase