Vector operations

Just like scalar information, vectors can also be added, subtracted and multiplied with each other. Suppose you have two vectors:  and  and  , with = (a_x, a_y, a_z) and  = (b_x, b_y, b_z). In this case, let us see how you will add and subtract these vectors with each other.

When adding the vectors, we add the components individually to give a new vector:

=  +  

= ((ax + bx) , (ay + by) , (az + bz ))

Let's now visualize the addition of two vectors in a graph as shown in the following image.

The Z value is kept as 0.0 for convenience:

Here, =(1.0, 0.4, 0.0 ) and  =(0.6, 2.0, 0.0)

So the resultant vector  =  +   = (1.0 + 0.6 , 0.4 + 2.0, 0.0 + 0.0)

                                                                      = (1.6, 2.4 , 0.0)

Vectors are also commutative meaning   +    will give the same result as  +  But if we add  to  then in the diagram above the dotted line will go from the tip of the blue arrow to the tip of the red arrow.

Furthermore, in vector subtraction, we subtract the individual components of the vectors to give a new vector:

=  -  

= ((ax - bx) , (ay - by) , (az - bz ))

Now let's visualize the subtraction of 2 vectors in a graph as shown in the following image:

Here   = (1.0, 0.4, 0.0 ) and  = (0.6, 2.0, 0.0)

So the resultant vector  =  -  = (1.0 - 0.6, 0.4 - 2.0, 0.0 - 0.0)

                                                                               = (0.4, -1.6, 0.0)

If vectors A and B are equal, the result will be a zero vector with all three components being zero.

If  = . This means a_x = b_x , a_y = b_y , a_z = b_z, then

=  -  = (0, 0, 0)

We can multiply a scalar with a vector. The result is again a vector with each component of the vector multiplied by the scalar.

For example, if A is multiplied by a single value of s we will have the following result:

 = s ×(a_x, a_y, a_z)

 = (s × a_x, s × a_y, s × a_z)

If  = (3, -5, 7) and s = 0.5 then  = s is

= (3 × 0.5, -5 × 0.5, 7 × 0.5)

= (1.5, -2.5. 3.5)