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In this lecture we will discuss multiplication of vectors. There are three ways to multiply vectors, the very first one is multiplication of a vector by a scalar, a scalar means by another number and then multiplication of two vectors such that the direction is no more required for the final form. The third one is multiplication of two vectors in which direction is required for the final form. Now as we know, a vector is actually a sum of magnitude plus direction. $$ Vector = Magnitude+Direction $$ When we multiply a vector by a scalar or a number, multiplication of this vector by a number will actually multiply with the magnitude while its direction will remain the same.
A vector can be any physical quantity; it may be a displacement, momentum or any other vector physical quantity. Let's say for example I'm having a vector which is in \(x\) direction with a magnitude 3. If I multiply this vector by 3, then $$ \vec { a } =3\hat { x } $$ $$ 3\vec { a } =3\times 3\hat { x } =9\hat { x } $$ When we multiply a vector by a number then it only changes the magnitude of the vector while its direction is not interrupted.
When we consider multiplication of two vectors, we will now deal not only with a magnitude but also with the direction. When two vectors are being multiplied then their respective magnitudes will multiply with each other but what will happen to their directions?
We will treat this in the vector algebra, which is very weird because in vector algebra two plus two may be or may not be four.
So, when two vectors are being multiplied their magnitudes will give us again the number but for their directions, there will be two possibilities. Either in the multiplication of two vectors we will get rid of the direction in the final form or we will have the direction.
So, the one that will have a direction, we will call it a vector product and the one which will have no direction, we will call that one as a scalar product.
Multiplication of vector in which there is no direction in the final form is called the dot product, scalar product or the inner product, which means that the inner components are multiplied in this case. It is represented by a dot “.” The result of this type of multiplication will be a scalar.
Example is work which is a dot product of two vec?tors, force and displacement. $$ W=\vec { F } .\vec { x } $$ This is the example of translational dynamics. And the other one is the cross product, this is also called vector product or the outer product, which is the opposite of inner product. It is represented by a cross “x”. The result of this will be a vector quantity. Examples are angular momentum or ?torque. $$ \tau =\vec { r } .\vec { F } $$ This is the example of rotational dynamics.
Whenever we will have a dot product it will mean that translation will take place, a body will move from one point to another point in a straight line. Whenever we will have a cross between vector quantities, rotation will be involved there.
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