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# Vector

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A vector is a mathematical object that has a size, called the magnitude, and a direction. It is often represented by boldface letters (such as $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}$), or as a line segment from one point to another (as in $\overrightarrow{AB}$).[1][2]

For example, a vector would be used to show the distance and direction something moved in. When asking for directions, if one says "Walk one kilometer towards the North", that would be a vector, but if they say "Walk one kilometer", without showing a direction, then that would be a scalar.

We usually draw vectors as arrows. The length of the arrow is proportional to the vector's magnitude. The direction in which the arrow points to is the vector's direction.[3]

## Examples of vectors

• John walks north 20 meters. The direction "north" together with the distance "20 meters" is a vector.
• An apple falls down at 10 meters per second. The direction "down" combined with the speed "10 meters per second" is a vector. This kind of vector is also called velocity.

## Examples of scalars

• The distance between two places is 10 kilometers. This distance is not a vector because it does not contain a direction.
• The number of fruit in a box is not a vector.
• A person pointing is not a vector because there is only a direction. There is no magnitude (the distance from the person's finger to a building, for example).
• The length of an object.
• A car drives at 100 kilometers per hour. This does not describe a vector, as there is only a magnitude, but no direction.

## More examples of vectors

• Displacement is a vector. Displacement is the distance that something moves in a certain direction. A measure of distance alone is a scalar.
• Force that includes direction is a vector.[3]
• Velocity is a vector, because it is a speed in a certain direction.[3][4]
• Acceleration is the rate of change of velocity. An object is accelerating if it is changing speed or changing direction.

## How to add vectors

### Adding vectors on paper using the head to tail method

Head-to-tail Addition

The Head to Tail method of adding vectors is useful for doing an estimate on paper of the result of adding two vectors. To do it:

• Each vector is drawn as an arrow with an amount of length behind it, where each unit of length on the paper represents a certain magnitude of the vector.
• Draw the next vector, with the tail(end) of the second vector at the head(front) of the first vector.
• Repeat for all further vectors: Draw the tail of the next vector at head of the previous one.
• Draw a line from the tail of the first vector to the head of the last vector - that's the resultant(sum) of all the vectors.

It's called the "Head to Tail" method, because each head from the previous vector leads in to the tail of the next one.

### Using component form

Using the component form to add two vectors literally means adding the components of the vectors to create a new vector.[5] For example, let a and b be two two-dimensional vectors. These vectors can be written in terms of their components.

$\mathbf{a} = ( a_x, a_y )$

$\mathbf{b} = ( b_x, b_y )$

Suppose c is the sum of these two vectors, so that c = a + b. This means that $\mathbf{c} = ( a_x + b_x, a_y + b_y )$.

Here is an example of addition of two vectors, using their component forms:

$\mathbf{a} = ( 3, -1 )$

$\mathbf{b} = ( 2, 2 )$

\begin{align}\mathbf{c} & = \mathbf{a} + \mathbf{b} \\ & = ( a_x + b_x, a_y + b_y ) \\ & = ( 3 + 2, -1 + 2 ) \\ & = ( 5, 1 ) \end{align}

This method works for all vectors, not just two dimensional ones.

## How to multiply vectors

### Using the dot product

The dot product is one method to multiply vectors. It produces a scalar. It uses component form:

\begin{align}\mathbf{a} & = (2, 3)\\ \mathbf{b} & = (1, 4) \\ \mathbf{a} \cdot \mathbf{b} & = (2,3)\cdot(1,4) \\ & = (2\cdot 1)+(3\cdot 4) \\ & = 2+12 \\ &= 14 \end{align}

### Using the cross product

The cross product is another method to multiply vectors. Unlike dot product, it produces a vector. Using component form:

$\mathbf{a}\times\mathbf{b}=|\mathbf{a}||\mathbf{b}|\sin(\theta) \mathbf{n}$

Here, $|\mathbf{a}|$ means the length of $\mathbf{a}$, and $\mathbf{n}$ is the unit vector at right angles to both $\mathbf{a}$ and $\mathbf{b}$.

### Multiplying by a scalar

To multiply a vector by a scalar (a normal number), you multiply the number by each component of the vector:

$c\,\mathbf{x} = (c\,x_1,c\,x_2,...,c\,x_n)$

An example of this is

\begin{align}c & = 5\\ \mathbf{x} & = (3,4)\\ c\,\mathbf{x} & = (5\cdot3, 5\cdot4)\\ & = (15, 20) \end{align}