Physics SCALAR AND VECTOR PRODUCTS
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SCALAR AND VECTOR PRODUCTS

Scalar Multiple of a Vector

When a vector is multiplied by a scalar `lamda`. we get a new vector which is `lamda` times the vector A i.e. `lamdavecA`. The direction of resulting vector is that of `vecA`.

If `lamda` has negative value, then we get a vector whose direction is opposite of `vecA`. The unit of resulting vector is the multiplied units of `lamda` and `vecA`.

For example, when mass is multiplied with velocity, we get momentum. The unit of momentum is obtained by multiplying units of mass and velocity.

Similarly, we can have vector A divided by a scalar `lamda`. The resulting vector becomes `(vecA)/lamda`.

The magnitude of the new vector becomes `1/lamda` that of `vecA` and direction is same as that of `vecA`.

Scalar Product or Dot Product

`vecA.vecB = |A|.|B| costheta`

where `theta` is the angle between the two vectors, when placed tail to tail.

`vecA.vecB` will be positive when angle between the vectors are acute.

`vecA.vecB` will be negative when angle between the vectors are obtuse.

`vecA.vecB` will be zero when angle between the vectors are `90^0`.

`text(Scalar Product or Dot Product In Unit Vectors Notation :)`

Let there be two vectors given by
`vecA =A_xhati +A_yhatj +A_zhatk` and `vecB =B_xhati +B_yhatj +B_zhatk`

`vecA.vecB=(A_xhati +A_yhatj +A_zhatk).(B_xhati +B_yhatj +B_zhatk)`

`=A_xB_x+A_yB_y+A_zB_z`

`text(Some properties of Dot product)`

It is commutative, i.e. `veca.vecb = vecb.veca`
It is distributive over addition and subtraction i.e. `veca.(vecbpmvecc)=veca.vecbpmveca.vecc`
If `veca` and `vecb` are perpendicular `veca.vecb = 0`
If `veca` and `vecb` are parallel `veca.vecb =ab`
Square of a vector is defined as `veca.veca=a^2`(scalar)
`hati.hati = hatj.hatj = hatk.hatk=1`
`hati.hatj = hatj.hatk = hatk.hati=0`

`text(Angle between two vectors :)`

As we know, `vecAvecB=|vecA|.|vecB|costheta`

`costheta=(vecA.vecB)/(|vecA|.|vecB|)=(A_xB_x+A_yB_y+A_zB_z)/(sqrt(A_x+A_y+A_z)sqrt(B_x+B_y+B_z))`

Vector Product or Cross product

The vector (or cross) product of two vectors `veca` and `vecb` is written as `vecaxxvecb` and is a vector `vecc` whose magnitude c is given by `c = ab sintheta` in which `theta` is the smaller of the angles between the direction of `veca` and `vecb`. The direction of `vecc` is perpendicular to the plane defined by `veca` and `vecb` and is given by right hand rule

`vecc = vecaxxvecb` (read as `veca` cross `vecb`)

`|vecc|=|vecaxxvecb|=ab sin theta`

and `vecaxxvecb=|veca||vecb|sintheta hatn`

where `hatn` is the unit vector

The vector `vecc` is directed perpendicular to the plane formed by `veca` and `vecb`. The direction of vector `vecc` may be obtained by using the Right Hand Thumb Rule. Stretch all the fingers and thumb of your right hand such that they are perpendicular to each other. Now align your hand such that its plane is perpendicular to the plane formed by vectors `veca` and `vecb`. Align the stretched fingers along the direction of the vector written first in order i.e., `veca` in this case. Curl the fingers of your hand towards the second vector (i.e. vector `vecb`) through the smaller angle. Then, the direction of the thumb gives the direction of the cross product.

`text(Properties of Cross Product)`

It is not commutative `vecaxxvecb ne vecbxxveca`. Infact `vecaxxvecb = -vecbxxveca`
It is distributive over addition and subtraction `vecaxx(vecb pm vecc)=vecaxxvecb pm vecaxxvecc`
If `veca||vecb`, then `vecaxxvecb=0` and if `veca bot vecb`, then `|veca xx vecb|=ab`
If `hati,hatj,hatk` be the unit vectors along in the positive directions of x, y, and z axes then
`hatixxhati=hatjxxhatj=hatkxxhatk=0`
`hatixxhatj=hatk`, `hatjxxhatk=hati`, `hatkxxhati=hatj`
The cross product may also be expressed by the determinant.

 
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