Scalar And Vector Products

Scalar products and vector products are 2 ways of multiplying 2 different vectors.

Scalar Product:

“Scalar products can be found by taking the component of one vector in the direction of the other vector and multiplying it with the magnitude of the other vector”. This can be represented as follows.

Scalar Product

Scalar product of \(\vec{A}.\vec{B}=ABcos\Theta\)

Where \(\vec{A}\) denotes the vector and \(\vec{A}\) denotes the magnitude of vector \(\vec{A}\), Scalar product is also termed as dot product  or inner product and remember that scalar multiplication is always denoted by a dot”.

If the same vectors are expressed in the form of unit vectors I, j and k along the axis x, y and z respectively, the scalar product can be expressed as follows,

\(\vec{A}.\vec{B}=A_{X}B_{X}+A_{Y}B_{Y}+A_{Z}B_{Z}\)

Where,

\(\vec{A}=A_{X}\vec{i}+A_{Y}\vec{j}+A_{Z}\vec{k}\)

\(\vec{B}=B_{X}\vec{i}+B_{Y}\vec{j}+B_{Z}\vec{k}\)

Matrix representation of scalar products,

It is useful to represent vectors as a row or column matrices, instead as above unit vectors. If we treat vectors as column matrices of their x, y and z components, then the transposes of these vectors would be row matrices. Therefore the vectors \(\vec{A}and\vec{B}\) would look like,

\(\vec{A^T}=A_XA_YA_Z\)

\(\vec{B}=\begin{bmatrix} B_X\\ B_Y\\ B_Z \end{bmatrix}\)

The matrix product of these 2 matrices will give us the scalar product of the 2 matrices which is the sum of corresponding spatial components of the given 2 vectors, the resulting number will be the scalar product of vector A and vector B.

\(\begin{bmatrix} A_X &A_Y &A_Z \end{bmatrix}\begin{bmatrix} B_X\\ B_Y\\ B_Z \end{bmatrix}=A_XB_X+A_YB_Y+A_ZB_Z=\vec{A}.\vec{B}\)

Vector Product:

The magnitude vector product of two given vectors can be found by taking the product of the magnitudes of the vectors times the sine of the angle between them. The magnitude of vector product can be represented as follows.

\(\vec{A}x\vec{B}=A\;BSin\Theta\)

Remember the above equation is only for the magnitude, for the direction of the vector product, following expression is used,

\(\vec{A}x\vec{B}=\vec{i}(A_YB_Z-A_ZB_Y)-\vec{j}(A_XB_Z-A_ZB_X)+\vec{k}(A_XB_Y-A_YB_X)\)

The above equation gives us the direction of the vector product,

Like scalar products, vector products can be represented by determinants,

\(\vec{A}x\vec{B}=\begin{vmatrix} \vec{i} &\vec{j} &\vec{k} \\ \vec{A_X}&\vec{A_Y} &\vec{A_Z} \\ \vec{B_X}&\vec{B_Y} &\vec{B_Z} \end{vmatrix}\)

Now the above determinant can be solved as follows,

\(\vec{A}x\vec{B}=\vec{i}(A_YB_Z-A_ZB_Y)-\vec{j}(A_XB_Z-A_ZB_X)+\vec{k}(A_XB_Y-A_YB_X)\)

Application of scalar and vector products are countless especially in situations where there are two forces acting on a body in a different direction, a good example would be the calculation of the magnetic force acting on a moving charge in a magnetic field, other applications include determining the net force on a body, etc.

Stay tuned with Byju’s to know more about scalar and vector products and much more.


Practise This Question

A force of 5 N acts on a particle along a direction making an angle of
60 with vertical. Its vertical component be