High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Linear Algebra Textbook > Vectors

Basics of Vectors

Vector Operations and Linear CombinationsNext Section
Popular Courses
Find My Course
0:00 / 0:00

Basics of Vectors

What is a vector?

A vector is a list of numbers that we denote v⃗=⟨v1,v2,...,vn⟩\vec{v} = \langle v_1, v_2, ..., v_n \rangle or v⃗=[v1v2⋮vn]\vec v = \begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n\end{bmatrix}.

The numbers v1, v2, ..., vnv_1,\ v_2,\ ...,\ v_n are called the components of the vector v⃗∈Rn\vec{v} \in \reals^n.
Geometrically, a vector is a directed line segment (an arrow).


Notes
  • We will write points with parentheses
  • e.g. Point A(1,2)A(1,2)
  • We will write vectors with angle brackets or as column vectors
  • e.g. Vector ⟨1,2⟩=[12]\lang 1,2\rang = \begin{bmatrix} 1\\ 2 \end{bmatrix}
Example 1
The vector ⟨1,0,−2⟩\lang 1,0,-2 \rang has three components, so it is in the space R3\reals^3:
  • xx-component:
    1
  • yy-component:
    0
  • zz-component:
    -2
We may write [10−2]∈R3\begin{bmatrix} 1\\ 0\\ -2 \end{bmatrix} \in \reals^3
PAGE BREAK

Scalars and Vectors

Scalars consist of a single number, a magnitude (size).
Think: temperature, dollars, perimeter

Vectors have both magnitude and direction.
Think: wind velocity, forces, displacement


PAGE BREAK

Position Vectors

The position vector of a point PP is the vector starting at the origin OO and ending at the point PP, denoted OP→\overrightarrow{OP}.
Position vectors are used to distinguish between points and vectors.


Example 2
What is the position vector of the point U(2,−3)U(2,-3)?

OU→=⟨2,−3⟩∈R2\overrightarrow{OU} = \lang 2,−3 \rang \in \reals^2

PAGE BREAK

Equal Vectors

Vectors with the same magnitude and in the same direction are equal.
  ⟹  \implies You are allowed to translate (move) a vector and it is still the same vector.

Furthermore, two vectors are equal if their corresponding components are all equal.

Watch Out!
Vectors from different spaces (different numbers of components) can never be equal!


Example 3
Even though the vector from AA to BB does not start at the origin, it has the same length and direction as OV→\overrightarrow{OV}.
Therefore, AB→=OV→\overrightarrow{AB} =\overrightarrow{OV} (they have the same position vector).


0:00 / 0:00

Example: Basics of Vectors

Express the position vector shown using two different notation styles.
How does this differ from the notation for the point VV?

V⃗=⟨4,−1,2⟩=[4−12]\vec V = \lang 4, -1, 2 \rang = \begin{bmatrix} 4\\ -1\\ 2\\ \end{bmatrix} and we write the point with parentheses: V(4,−1,2)V(4,-1,2)

In what space does this vector lie?
V⃗∈R3\vec V \in \reals^3
0:00 / 0:00

Example: Equal Vectors

Examine the vector below. What are the components of the equivalent vector that starts at the origin (the position vector)?


To move the tail of the vector to the origin, we can move the entire vector left 1 and up 1.
This would move the tip/head of the vector to the point (−1,3)(-1, 3).

Therefore, the position vector [−13]\begin{bmatrix} -1\\ 3\\ \end{bmatrix} is equal to the vector seen above.

Practice: Basics of Vectors

Examine the following graph.






Determine the components of the vector shown.
Extra Practice
Basics of Vectors
Practice: Basics of Vectors
Examine the following graph.
Find the vector u⃗\vec{u} which has the opposite direction as v⃗=[1−82]T\vec{v}=\begin{bmatrix}1&-8&2\end{bmatrix}^T but half the length.
Find the vector u⃗\vec{u} which has the opposite direction as v⃗=[1−82]T\vec{v}=\begin{bmatrix}1&-8&2\end{bmatrix}^T but half the length.
Given the points A(0,−1,3)A\left(0,-1,3\right) and B(5,−6,−2)B\left(5,-6,-2\right), find BA⃗+AB⃗\vec{BA}+\vec{AB}.
*Type any vector answers in the form (a, b, c)
special unit vector concept clarifier
Wize Concept Clarifier:
Express 𝑣⃗=(15,0,−2)\vec{𝑣} = (15, 0, −2) in terms of special unit vectors
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
Concept Clarifier
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
Practice: Scalar or Vector?
Each of the following is either a scalar or a vector quantity.
Select all of the vector quantities in the list.
Fill in the blanks.
A vector quantity has both a ____[A]____ and a ____[B]______ while while scalar quantities have only a _____[C]______
Practice Question: Direction Vector
Practice Question: Direction Vector
Which of the following vectors is not parallel to the line passing through the points (1,2) and (−3,4)?(1,2)\ and\ (−3,4)?
a.) (5, −52)\left(5,\ -\frac{5}{2}\right)
Practice: Vector Properties
Given the points A(1, −3,2)A\left(1,\ -3,2\right), B(0, 1, 1)B\left(0,\ 1,\ 1\right), C(4, −15, 5)C\left(4,\ -15,\ 5\right), wich of the following statements is/are true?
(Select all that apply)
Practice: Vector Operations
A parallelogram has sides ABAB, BCBC, CDCD and DADA
The following coordinates are known: A(2,0,1)A(2,0,1), C(6,0,0)C(6,0,0), and M(3,1,0)M(3,1,0), where MM is the midpoint between AA and BB
Find the vector BD⃗\vec{BD}
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ | Proofed. 19.1W
The vector going from the point P1(−1,1,−4)P_1(-1,1,-4) to the point P2(1,4,2)P_2(1,4,2) is:
Practice: 2D Vectors
Practice: 2D Vectors
Given the diagram below, answer the following questions.