Components of vectors





Components of a vector:

The components of a vector are two or more vectors ( Usually vectors along the x , y , z axes) whose vector sum is equal to the given vector.

For ease in doing vector calculations Vectors are expressed in the form of two or three components ; Two if the vector is two dimensional and three if the vector is three dimensional .

Components of a Two dimensional vector:

Consider a two dimensional vector \overrightarrow{a} whose initial point is the point “o” the rectangular coordinate system and final point “A”.

If we produce the lines from “O” and “A” which meets at “C” making 90 degree angle with each other , then the two newly formed vectors \overrightarrow{a_x} and \overrightarrow{a_y} are called the components of vector \overrightarrow{a} , as shown in the figure below:

components of vectors

components of vectors

 

Components of a three dimensional vector:

 

Similar to two dimensional components of a vector if we resolve a vector \overrightarrow{a} into it’s components \overrightarrow{a_x} , \overrightarrow{a_y} and \overrightarrow{a}_z in the rectangular coordinate system ( x , y and z axes or three dimensions ) then the newly formed three vectors are called the x , y and z components of vector \overrightarrow{a} respectively , as shown in the figure below:

components of vectors

components of vectors

 

 

Resolving a Vector into it’s components:

We can resolve a vector into it’s components if enough data about the vector is given to us.

To resolve a vector into it’s components we can use the formulas below:

To resolve two dimensional vector:

If ,  a is the magnitude of vector \overrightarrow{a} and  \theta is the angle made by the vector with x axes or the direction of the vector then:

a_x = a \times \cos \theta

and a_y = a \times \sin \theta

where ,  a_x is the magnitude of x component of vector \overrightarrow{a} and   a_y is the magnitude of y component of vector \overrightarrow{a}

a_x is a vector along x-axes and a_y is a vector along y-axes.

 

To resolve a three dimensional vector:

If , If ,  a is the magnitude of vector \overrightarrow{a} ,  \theta is the angle made by the vector with x axes  and  \phi is the angle made by the vector with x-y plane. then:

 

a_x = a \times \cos \theta

,

a_y = a \times \sin \theta

and a_z = \sqrt{a_x ^2 + a_y^2} \times \sin \phi

where ,  a_x is the magnitude of x component of vector \overrightarrow{a}  a_y is the magnitude of y component of vector \overrightarrow{a} and   a_z is the magnitude of z component of vector \overrightarrow{a}.

a_x is a vector along x-axes , a_y is a vector along y-axes and a_z is a vector along z-axes.

 

 

Unit Vector Notation:

A vector can be denoted by using the concept of unit vector and components of vector in the unit vector form as following:

\overrightarrow{a} = a_x . \hat{i} + a_y . \hat{j} + a_z . \hat{k}

where ,  a_x is the magnitude of x component of vector \overrightarrow{a}   a_y is the magnitude of y component of vector \overrightarrow{a} and   a_z is the magnitude of z component of vector \overrightarrow{a}.



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