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Two ways to multiply two vectors are vector products and scale products. They have the greatest applications in astronomy and physics. The product of the magnitudes of both vectors and the cosine between them is called the scalar product.
Scalar Product
Scalar products are obtained by multiplying the magnitude of the other and taking the component in one vector’s direction. It can be described as:
Scalar product, also known as dot product, is an algebraic operation that takes two sequences of numbers of equal length and returns one number.
It can be written as follows:
Scalar product of A.B = ABcosθ
Where,
- A denotes the vector
- A denotes the magnitude of vector A
The scalar product can also be called the dot product, inner product, and you should remember that scalar multiplication is always indicated by a dot.
The scalar product is expressed if the same vectors can be expressed in unit vectors I. j. and k, respectively, along the axis of x,y, and z.
A.B=AxBx+AyBy+AzBz
Where,
A=Axi+Ayj+Azk
B=Bxi+Byj+Bzk
Matrix Representation of Scalar Products
It is more useful to think of vectors as column or row matrices than as above-unit vectors. If vectors are treated as column matrices for their x,y, and z components then the transposes would be row matrices. The vectors A and B would then look like this:
A=[Ax Ay Az ]
B= Bx By Bz
These 2 matrices are multiplied to give us the matrix product. This is the sum of the spatial components of each vector. The resulting number will be vector A and vector B.
AxAyAz Bx By Bz = AxBx+AyBy+AzBz = AB
Vector Product
You can find the magnitude vector product between two vectors by multiplying the magnitudes of each vector and the sine of their angle. It can be described as:
Cross product, also known as vector product, is a binary operation that takes two vectors and applies them in three-dimensional space.
Here are some ways to represent the magnitude of a vector product:
AxB =ABsin
The above equation only represents the magnitude. For the direction of vector products, the following expression should be used.
AxB =iAyBz-AzBy-jAxBz-AzBx-kAxBy-AyBx
[The vector product direction is given by the above equation]
Determinants Represent Vector Products
AxB = i j k ,Ax Ay Az , Bx By Bz
The above determinant can now be solved by the following:
AxB =iAyBz-AzBy-jAxBz-AzBx-kAxBy-AyBx
There are many ways to apply vector and scalar products, especially when two forces are acting in different directions on a body.
