Vector Multiplication

1. Scalar Multiplication

Scalar Multiplication

Multiplying a given vector say A by a scalar say m generates a new vector say B which is called  as a scalar multiple of A. The vectors which are lying along the same straight line are known as  co-linear vectors.  

A = x î + y ĵ

when A is multiplied by the scalar a then       

B=m A = m(x î + y ĵ)

Multiplication of a vector by a scalar is distributive.

2. Dot Product

Consider two vectors a and b, the angle between them being α. For the sake of simplicity we have represented them in two dimensions in the figure below:

The dot product or scalar product returns a scalar (that is, a number) and is given by:

From the expression above we can see that the dot product of two perpendicular vectors is 0.

The dot product can be used to find the projection of a vector onto another one. As an example we are going to find the projection of a onto b.

The projection of a vector on a line can be found by drawing a line from the end of the vector perpendicular to the line onto which we want to project it. The resulting distance over the line is the projection a_b:

Analytically it can be found by using the cosine function:

The direction defined by a vector is given by the unit vector u_b:

Which is precisely the projection of a onto b.

The dot product is commutative:

From the previous results, it can be deduced:

When both vectors a and b are expressed in unit vector notation, as shown in the first figure, the dot product is given by:

Finally, we can find the angle between two vectors by using the dot product:

The dot product is used in Physics to define the work of a force.

3. Cross Product

The cross product or vector product of two vectors a ⨯ b is a vector that is perpendicular to both a and b whose direction is given by the right-hand rule:

Simply points the palm of the right hand in the direction of a and closes it in the direction of b. The direction of the cross product of both vectors is coming out of the thumb.

The magnitude of the cross product is given by:

From the previous expression it can be deduced that the cross product of two parallel vectors is 0.

The cross product is anti-commutative; if we apply the right-hand rule to multiply b ⨯ a it gives:

This vector has the same magnitude as a ⨯ b, but points in the opposite direction. And two vectors are equal only if they have both the same magnitude and direction.

When two vectors are given in unit vector notation , their cross product is given by the following determinant:

And expanding the previous expression:

The magnitude of the cross product is equal to the area of the parallelogram that the vectors span. Viewing both from above:

The area of a parallelogram is its base multiplied by its height, therefore:

4. Problem Questions

Try to solve this problems.. No need to submit 

Given the vectors: A = 3i + 2j – k and B = 5i +5j, find:

1. The dot product A⋅B.
2. The projection of A onto B.
3. The angle between A and B.
4. A vector of magnitude 2 in the XY plane perpendicular to B.