Notation and Terminology
In a vector,
is the start point and
is the endpoint. And, the magnitude of such a vector is
. Equal vectors have the same direction and magnitude.
The negative of a vector, , has the same magnitude, but in the opposite direction of
.
Addition and Subtraction
To add vectors in a coordinate plane, we have:
(1)
![](https://www.mohsenhamedi.com/wp-content/uploads/2021/10/addition-300x243.png)
And the subtraction of two vectors is
(2)
This is equivalent to
(3)
which is depicted in figure a.
![](https://www.mohsenhamedi.com/wp-content/uploads/2021/10/subtraction.png)
Magnitude of a Vector
The magnitude, length, or norm of a vector is
(4)
A unit vector, is a vector that has a magnitude of
![Rendered by QuickLaTeX.com 1](https://www.mohsenhamedi.com/wp-content/ql-cache/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png)
Normalization of a Vector
The normalized vector of is a unit vector in the same direction as
. This means that a non-zero vector
is normalized as follows
(5)
where
![Rendered by QuickLaTeX.com \boldsymbol{u}](https://www.mohsenhamedi.com/wp-content/ql-cache/quicklatex.com-152df8a762168f7a0ff6cf86e9405136_l3.png)
![Rendered by QuickLaTeX.com \boldsymbol{a}](https://www.mohsenhamedi.com/wp-content/ql-cache/quicklatex.com-1c5eb72f07e5663850930f85eeb90384_l3.png)
The Unit Vectors in 2 Dimensions
The unit vectors and
are known as the
and
vectors, respectively.
In space,
and
are the horizontal and vertical components of
, respectively.
The Unit Vectors in 3 Dimensions
The unit vectors , and
are known as the
, and
vectors, respectively.
The Distance Between Two Points
The distance between and
in 3-space is
(6)
Midpoint Formula
The midpoint of line segment between points and
is
(7)
Vectors in 3-Space
Similar to space, vectors are defined in
space as
.
Reference
Dennis G. Zill. Advanced Engineering Mathematics, edition. Jones
Bartlett Learning. 2016.