2.1: Vectors

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Vectors are fundamental objects in applied mathematics; they efficiently convey information about a mathematical or physical object. Let’s get a sense of what they are.

Definition: Vector

A VECTOR is a representation of an object that has both direction and magnitude. By direction, we mean the place toward which something faces, and by magnitude, we mean the size of something.

A vector can be depicted visually by an arrow, with an initial point called the tail and a terminal point called the head. The length of the arrow represents the vector’s magnitude.

$This image shows the different aspects of the vector. There is the direction of the vector, which is pointing somewhat in the northeast direction. There is the magnitude, which is the length of the vector. There is the tail, which is the initial or starting point of the vector. Finally, there is the head, which is the terminal or ending point of the vector.$

$clipboard_eb82b33649f634945d39a0f25228edf40.png$

An example of a vector is a car’s velocity. Velocity is a vector since it has both magnitude (speed) and direction. A car might be moving west at 60 mph. Other examples of vectors are displacement, acceleration, and force.

The temperature of some medium is not a vector since it has only magnitude. But if the medium is being heated, its temperature is increasing and has a direction; it is going upward. The increase or decrease in temperature is a vector.

Vectors in Standard Position

$Image of a Cartesian coordinate system with two vectors in a standard position. Vector u is pointing southwest in the third quadrant, where x and y values are both negative. Vector v is pointing northeast in the third quadrant, where x and y values are both positive.$

$clipboard_e09350968f810c666c50c15eb022cb97f.png$

Components of a Vector

$clipboard_e91b670d0b2d62b35c4f50f9ffec38202.png$

$Diagram image of vector v pointing in the northeast direction.$

For example, the vector $\vec{v}$ in the diagram can be broken into two components,

its horizontal, or $x$-component, and
its vertical, or $y$-component.

$This image shows the different directional components of vector v which we call the x-component and the y-component. The x-component points left or right and the y-component points up or down. The x-value and the y-value shows how many units it points in that direction. This vector is 3 units long to the left and 6 units long up, so the x-value is 3 and the y-value is 6.$

The vector $\vec{v}$ in component form is expressed using angle brackets as $\vec{v}=\langle 3,6\rangle$, where

the first component, 3 is the length and direction of its $x$-component, and
the second component, 6 is the length and direction of its $y$-component.

The vector $\overrightarrow{u}$ in the picture below has

FIRST COMPONENT = (terminal $x$-value) – (initial $x$-value) = $2-7=-5$, and

SECOND COMPONENT = (terminal $y$-value) – (initial $y$-value) = $4-6=-2$,

so that $\overrightarrow{u}=\left\langle -5,\left.-2\right\rangle \right.$.

This image shows the different directional components of vector u. If the directional component of the x-value is left, then the x-value is negative. If the directional component of the y-value is down, the y-value is negative. Since the vector points 5 units to the left and 2 units down, the x-value is negative 5 and the y-value is negative 2. For <x,y we get <-5,-2>." src="/@api/deki/files/97342/clipboard_eaa414d12e539b2aa26d4cc28d0dc560e.png">

Row and Column forms of a Vector

Vectors are represented by a single column matrix or a single row matrix. The vectors $\overrightarrow{v}=\left\langle 3,\left.6\right\rangle \right.$, and

$\overrightarrow{u}=\left\langle -5,\left.-2\right\rangle \right.$ above, can be represented by the 2x1 row matrix and the 1x2 column matrix, respectively as

\[\vec{v}=\left[\begin{array}{ll}
3 & 6
\end{array}\right] \text { and } \vec{u}=\left[\begin{array}{l}
-5 \\
-2
\end{array}\right] \nonumber \]

Equal Vectors

Definition: Equal Vectors

Two vectors are EQUAL if they have the same direction and magnitude. They may start and end at different positions, but their representing arrows will be parallel.

$This image shows two vectors a and b pointing in the exact same direction. However, vector b is 6 units above vector a. The initial point for vector b is (6,3) and the terminal point is (3,8.5). The initial point for vector a is (6, negative 3) and the terminal point is (3,2.5).$

In the diagram vectors $\vec{a}$ and $\vec{b}$ are equal but appear in different locations in the x-yplane.

Try These

Exercise $\PageIndex{1}$

Express the vectors $\vec{v}$ and $\vec{u}$ in component form.

$This image shows two different vector u and v. Vector u is longer than vector v and both are pointing horizontally in opposite directions and the same direction vertically. The initial point vector u is (negative 5,2) and the terminal point is (8,5). The initial point for vector v is (negative 2, negative 8) and the terminal point is (negative 5, negative 6).$

Answer: $\overrightarrow{v}=\left\langle -3,\left.2\right\rangle \right.$ and $\overrightarrow{u}=\left\langle 13,\left.3\right\rangle \right.$

Exercise $\PageIndex{2}$

Explain why the two vectors are equal.

$These are two vectors pointing in the same direction but at different positions. The starting point for the top vector is (negative 4,8) and the terminal point is (4,4). The starting point for the top vector is ( negative 1,0) and the terminal point (7, negative 4).$

Answer: Two vectors are equal because they have the same direction and magnitude.

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Definition: Equal Vectors

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)