# Vectors Operations

If we have two vectors,

and we want to add them, we use the formula:

Similarly, if we want to subtract them:

As a numerical example consider the vectors:

The result would be:

### Multiplying a Scalar by a Vector

To multiply the scalar k by the vector:

We multiply the scalar by all the elements of the vector. The result is the initial vector scaled by k:

As a numerical example consider k=2 and the vector:

The result is:

Image 2: Multiplying a Scalar by a Vector

### Multiplying Vectors

1) Dot Product or Internal Product of Vectors

The dot product of a vector by another vector results in a scalar. Consider the following vectors:

If the angle θ between the vectors is known, the dot product can be calculated by the relationship:

Geometrically, the dot product of two vectors is the product of the magnitude of the fist vector by the projection of the second, on the first.

Image 3: Dot Product of two Vectors

Dot product expresses how much a vector acts on another vector. If a vector is vertical to another vector, then there is no action. If a vector is in line with another vector then the action is maximum. For example if an object can be moved only horizontally and you apply a vertical force on it, then there is no action. But if you apply the force horizontally then all the force will produce action. If the force is applied by an angle, then only the horizontal component of the force will be useful.

The dot product is commutative and distributive:

Applying the dot product to the unit vectors we get:

From the equations above we can calculate the following:

The last equation is very useful in calculating the dot product.

The dot product of a vector by itself is equal to the square of its magnitude:

Or:

2) Cross Product or External Product of Vectors

The cross product of a vector by another vector results in a new vector which is perpendicular to the plane that is defined by the two initials and its direction follows the right hand rule. Consider the following vectors:

The cross product is:

Also if the angle θ between the vectors is known, the cross product can be calculated by the relationship:

Where the vector

is perpendicular to the plane that is defined by the vectors

Geometrically the magnitude of the cross product of two vectors is the area of the parallelogram that is formed between them.

Image 4: Cross Product

The dot product is not commutative, so mind the sequence of the vectors in the product, but it is distributive:

Applying the cross product to the unit vectors we get:

The cross product of a vector by itself is equal to zero:

From the equations above we can calculate the following:

The determinant form of the cross product is the main formula used to calculate cross products.

3) Triple Vector Products

There are some triple products that are quite useful.