Vectors Examples

• Addition and Subtraction of Vectors
  
  You can add or subtract two vectors. Both the operand vectors must be of same type and have same number of elements.
  

  Example

  Create a script file with the following code:
  

  A = [7, 11, 15, 23, 9];
  B = [2, 5, 13, 16, 20];
  C = A + B;
  D = A - B;
  disp(C);
  disp(D);

  When you run the file, it displays the following result:
  

  9    16    28    39    29
  5     6     2     7   -11 

• Scalar Multiplication of Vectors
  
  When you multiply a vector by a number, this is called the scalar multiplication. Scalar multiplication produces a new vector of same type with each element of the original vector multiplied by the number.
  

  Example

  Create a script file with the following code:
  

  v = [ 12 34 10 8];
  m = 5 * v

  When you run the file, it displays the following result:
  

  m =
      60   170    50    40
  

  Please note that you can perform all scalar operations on vectors. For example, you can add, subtract and divide a vector with a scalar quantity.
  
• Transpose of a Vector
  
  The transpose operation changes a column vector into a row vector and vice versa. The transpose operation is represented by a single quote(').
  

  Example

  Create a script file with the following code:
  

  r = [ 1 2 3 4 ];
  tr = r';
  v = [1;2;3;4];
  tv = v';
  disp(tr); disp(tv);

  When you run the file, it displays the following result:
  

       1
       2
       3
       4
  
       1     2     3     4 

• Appending Vectors
  
  MATLAB allows you to append vectors together to create new vectors.
  If you have two row vectors r1 and r2 with n and m number of elements, to create a row vector r of n plus m elements, by appending these vectors, you write:
  

  r = [r1,r2]

  You can also create a matrix r by appending these two vectors, the vector r2, will be the second row of the matrix:
  

  r = [r1;r2]

  However, to do this, both the vectors should have same number of elements.
  Similarly, you can append two column vectors c1 and c2 with n and m number of elements. To create a column vector c of n plus m elements, by appending these vectors, you write:
  

  c = [c1; c2]

  You can also create a matrix c by appending these two vectors; the vector c2 will be the second column of the matrix:
  

  c = [c1, c2]

  However, to do this, both the vectors should have same number of elements.
  

  Example

  Create a script file with the following code:
  

  r1 = [ 1 2 3 4 ];
  r2 = [5 6 7 8 ];
  r = [r1,r2]
  rMat = [r1;r2]
   
  c1 = [ 1; 2; 3; 4 ];
  c2 = [5; 6; 7; 8 ];
  c = [c1; c2]
  cMat = [c1,c2]

  When you run the file, it displays the following result:
  

  r =
       1     2     3     4     5     6     7     8
  rMat =
       1     2     3     4
       5     6     7     8
  c =
       1
       2
       3
       4
       5
       6
       7
       8
  cMat =
       1     5
       2     6
       3     7
       4     8 

• Magnitude of a Vector
  
  Magnitude of a vector v with elements v1, v2, v3, …, vn, is given by the equation:
  |v| = √(v1² + v2² + v3² + … + vn²)
  You need to take the following steps to calculate the magnitude of a vector:
  
• Take the product of the vector with itself, using array multiplication (.*). This produces a vector sv, whose elements are squares of the elements of vector v.
  sv = v.*v;
• Use the sum function to get the sum of squares of elements of vector v. This is also called the dot product of vector v.
  dp= sum(sv);
• Use the sqrt function to get the square root of the sum which is also the magnitude of the vector v.
  mag = sqrt(s);

Example

Create a script file with the following code:

v = [1: 2: 20];
sv = v.* v;     %the vector with elements 
                % as square of v's elements
dp = sum(sv);    % sum of squares -- the dot product
mag = sqrt(dp);  % magnitude
disp('Magnitude:'); disp(mag);

When you run the file, it displays the following result:

Magnitude:
   36.4692  

• Vector Dot Product
  
  Dot product of two vectors a = (a1, a2, …, an) and b = (b1, b2, …, bn) is given by:
  a.b = ∑(ai.bi)
  Dot product of two vectors a and b is calculated using the dot function.
  

  dot(a, b);

  Example

  Create a script file with the following code:
  

  v1 = [2 3 4];
  v2 = [1 2 3];
  dp = dot(v1, v2);
  disp('Dot Product:'); disp(dp);

  When you run the file, it displays the following result:
  

  Dot Product:
      20 

• Vectors with Uniformly Spaced Elements
  
  MATLAB allows you to create a vector with uniformly spaced elements.
  To create a vector v with the first element f, last element l, and the difference between elements is any real number n, we write:
  

  v = [f : n : l]

  Example

  Create a script file with the following code:
  

  v = [1: 2: 20];
  sqv = v.^2;
  disp(v);disp(sqv);

  When you run the file, it displays the following result:
  

  1     3     5     7     9    11    13    15    17    19
  1     9    25    49    81   121   169   225   289   361