Unit Normal Vector

Vector function is different from scalar function because the vector function have the scalar value and direction normal vector is defined as the vector normal or perpendicular to the surface also unit vector is a vector whose length is 1 the unit normal vector is a vector perpendicular to the surface with a length value of 1 it is denoted by “hat” n.

Unit Normal Vector Formula

Below is the formula for unit normal vector:

The unit normal vector for the plane is` N = (dT)/(dphi)`

where dT= unit tangent vector

`dphi` = polar angle.

Unit tangent vector with ,

the normal vector

or

`N(t) =(T'(t))/(||T'(t)||)`

where `N(t)` = unit normal vector

`T(t)` = unit tangent vector = `(v(t))/(||v(t)||)`

Where v(t) = velocity vector = r’(t)

r(t) = differentiable vector

Solved Examples

Below is an example for unit normal vector:

Example: Find the unit normal vector for the vector valued function r(t) = t²i + t³ j .

Solution: Differentiable vector r(t) = t²i + t³ j

v(t) = r’(t)= `(d(t^2i + t^3j ))/(dt)`

v(t) = `2t i + 3 t^2j`

modulus of ¦v(t) ¦= `sqrt(4t^2+9t^4)`

= `sqrt(4t^2+9t^4)`

Therefore unit tangent vector T(t) = (v(t))/(¦v(t) ¦)

=  `(2t i + 3 t^2j)/(sqrt(4t^2+9t^4))`

T'(t) = `( [(2t i + 3 t^2j)(1/2) (8t+36t^3)(4t^2+9t^4)^(-1/2)-(4t^2+9t^4)^(1/2)(2i + 6t j)]) /(((4t^2+9t^4)^(1/2))^2)`

`= ((1/2)(16t^2i + 24t^3j+72t^4 i+108t^5j) (4t^2+9t^4)^(-1/2)-(4t^2+9t^4]^(1/2)(2i + 6t j))/ (4t^2+9t^4)`

=`(8t^2i + 12t^3j+36t^4 i+54t^5j) /((4t^2+9^4)^(1/2))-((4t^2+9t^4)^(1/2)(2i + 6t j))/ (4t^2+9t^4)`

=`((8t^2+36t^4)i + (12t^3+54t^5)j)/((4t^2+9t^4)^(3/2))-(2i + 6t j)/((4t^2+9t^4)^(1/2)) `

`=((8t^2+36t^4)i + (12t^3+54t^5)j)/((4t^2+9t^4)^(3/2))-((2 i+6t j)( 4t^2+9t^4))/((4t^2+9t^4)^(3/2)) `

`=((8t^2+36t^4)i + (12t^3+54t^5)j)/((4t^2+9t^4)^(3/2))-((8t^2+18t^4)i+(24t^3+54t^5)j)/((4t^2+9t^4)^(3/2)) `

=`((8t^2+36t^4-8t^2-18t^4)i + (12t^3+54t^5-24t^3+54t^5)j)/((4t^2+9t^4)^(3/2)) ` 

T’(t)= (18t^4 i + (-12t^3)j)/((4t^2+9t^4)^(3/2))

||T'(t)|| = `sqrt(((18 t^4)^2+(-12t^3)^2)/((4t^2+9t^4)^(3/2))) `

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Therefore the unit normal vector    

N(t) =(T(t))/(||T'(t)||)

By substituting T'(t) and ||T'(t)|| value in N(t) we get,

`= (((18t^4i + (-12t^3)j))/((4t^2+9t^4)^(3/2)))/(((18 t^4)^2 + (-12t^3)^2)/(sqrt((4t^2+9t^4)^3)))`

`=((18t^4i -12t^3j)((4t^2+9t^4)^(3/2)))/(((4t^2+9t^4)^(3/2))((324t^8-144t^6)^(1/2)))`

`=(18t^4i -12t^3j)/((324t^8-144t^6)^(1/2)) `

By simplifying above equation we get N(t)` = (9t i-6j)/ (sqrt(9t^2-4)) `

Therefore the unit normal vector for the vector valued function r(t) = t²i + t³ j is

N(t)=`(9ti-6j)/ (sqrt(9t^2-4))`