# MATLAB Programming/Arrays/Basic vector operations

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A vector in MATLAB is defined as an array which has only one dimension with a size greater than one. For example, the array [1,2,3] counts as a vector. There are several operations you can perform with vectors which don't make a lot of sense with other arrays such as matrices. However, since a vector is a special case of a matrix, any matrix functions can also be performed on vectors as well provided that the operation makes sense mathematically (for instance, you can matrix-multiply a vertical and a horizontal vector). This section focuses on the operations that can only be performed with vectors.

## Declaring a vector

Declare vectors as if they were normal arrays, all dimensions except for one must have length 1. It does not matter if the array is vertical or horizontal. For instance, both of the following are vectors:

```>> Horiz = [1,2,3];
>> Vert = [4;5;6];
```

You can use the isvector function to determine in the midst of a program if a variable is a vector or not before attempting to use it for a vector operation. This is useful for error checking.

```>> isvector(Horiz)
ans = 1
>> isvector(Vert)
ans = 1
```

Another way to create a vector is to assign a single row or column of a matrix to another variable:

```>> A = [1,2,3;4,5,6];
>> Vec = A(1,:)
Vec = 1   2   3
```

This is a useful way to store multiple vectors and then extract them when you need to use them. For example, gradients can be stored in the form of the Jacobian (which is how the symbolic math toolbox will return the derivative of a multiple variable function) and extracted as needed to find the magnitude of the derivative of a specific function in a system.

### Declaring a vector with linear or logarithmic spacing

Suppose you wish to declare a vector which varies linearly between two endpoints. For example, the vector [1,2,3] varies linearly between 1 and 3, and the vector [1,1.1,1.2,1.3,...,2.9,3] also varies linearly between 1 and 3. To avoid having to type out all those terms, MATLAB comes with a convenient function called linspace to declare such vectors automatically:

```>> LinVector = linspace(1,3,21)
LinVector = Columns 1 through 9
1.0000    1.1000    1.2000    1.3000    1.4000    1.5000    1.6000    1.7000    1.8000
Columns 10 through 18
1.9000    2.0000    2.1000    2.2000    2.3000    2.4000    2.5000    2.6000    2.7000
Columns 19 through 21
2.8000    2.9000    3.0000
```

Note that linspace produces a row vector, not a column vector. To get a column vector use the transpose operator (') on LinVector.

The third argument to the function is the total size of the vector you want, which will include the first two arguments as endpoints and n - 2 other points in between. If you omit the third argument, MATLAB assumes you want the array to have 100 elements.

If, instead, you want the spacing to be logarithmic, use the logspace function. This function, unlike the linspace function, does not find n - 2 points between the first two arguments a and b. Instead it finds n-2 points between 10^a and 10^b as follows:

```>> LogVector = logspace(1,3,21)
LogVector = 1.0e+003 *
Columns 1 through 9
0.0100    0.0126    0.0158    0.0200    0.0251    0.0316    0.0398    0.0501    0.0631
Columns 10 through 18
0.0794    0.1000    0.1259    0.1585    0.1995    0.2512    0.3162    0.3981    0.5012
Columns 19 through 21
0.6310    0.7943    1.0000
```

Both of these functions are useful for generating points that you wish to evaluate another function at, for plotting purposes on rectangular and logarithmic axes respectively.

## Vector Magnitude

The magnitude of a vector can be found using the norm function:

```>> Magnitude = norm(inputvector,2);
```

For example:

```>> magHoriz = norm(Horiz)
magHoriz = 3.7417
>> magVert = norm(Vert)
magVert = 8.7750
```

The input vector can be either horizontal or vertical.

## Dot product

The dot product of two vectors of the same size (vertical or horizontal, it doesn't matter as long as the long axis is the same length) is found using the dot function as follows:

```>> DP = dot(Horiz, Vert)
DP = 32
```

The dot product produces a scalar value, which can be used to find the angle if used in combination with the magnitudes of the two vectors as follows:

```>> theta = acos(DP/(magHoriz*magVert));
>> theta = 0.2257
```

Note that this angle is in radians, not degrees.

## Cross Product

The cross product of two vectors of size 3 is computed using the 'cross' function:

```>> CP = cross(Horiz, Vert)
CP = -3   6   -3
```

Note that the cross product is a vector. Analogous to the dot product, the angle between two vectors can also be found using the cross product's magnitude:

```>> CPMag = norm(CP);
>> theta = asin(CPMag/(magHoriz*magVert))
theta = 0.2257
```

The cross product itself is always perpendicular to both of the two initial vectors. If the cross product is zero then the two original vectors were parallel to each other.