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Wize High School Algebra II Textbook (Common Core) > Vectors (Extension Topic)

Basics of Vectors

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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
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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


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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

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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).


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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
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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.