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.
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