Unit Vector Formula

To understand the concept of the unit vector formula in physics, one must be clear regarding what a vector is.

A vector is a quantity that has magnitude along with the direction.

A vector is defined numerically by an arrow over the letter. And the vector's value is always represented by the mod.

To comprehend the unit vector formula, everyone must first actually recognize what a unit vector is.

[Image will be Uploaded Soon]

Unit Tangent Vector Formula

Suppose,

Given, function f(x)=x3+x+x2 which is differentiable in x.

Therefore, f'(x)=(3x2) +1+2x

f'(x) is called the velocity vector.

The tangent vector equation is then the unit vector in the form of the velocity vector and is used by the unit tangent vector equation to calculate the vector's length.

now,

||f'(x)||=√[(9x4) +1+4x2]

Then,

unit tangent vector T(x)=f'(x)/||f'(x)||.... (3)

= ((3x2) +1+2x)/√[(9x4) +1+4x2]

Principal Unit Normal Vector Formula

The normal vector to a surface, also generally known as the "normal," is a vector that is orthogonal to the surface at a given point.

There is an infinite number of perpendicular vectors to a vector in space. Our primary objective is to select a special vector right angles to the unit tangent vector.

Let f(x) be a differentiable vector-valued function and let T(x) be the unit tangent vector. Then,

where from equation (3),

T(x)=f'(x)/||f'(x)||

Therefore,

the unit normal vector formula is, N(x) =T′(x)/||T′(x)||

As directly compared to the formula for the unit tangent vector, when we consider the unit tangent vector as a vector-valued function, then the principal unit normal vector N(x) is the unit tangent vector of the unit tangent vector function.

Solved Examples

Example 1:

Suppose,

There is a vector a= (1,0).

To find the magnitude of the given vector, we use the formula,

|a|= √(x2+y2+z2) ......(1)

Using the magnitude of unit vector formula (1),

|a|= √ (12+0) =√1=1

Therefore, a is a unit vector.

Example 2:

Suppose,

There is a vector b= (2, 3).

To find the magnitude of the given vector, we use the formula,

|b|= √(x2+y2+z2) ......(1)

Using the magnitude of unit vector formula (1),

|b|= √ (22+32)

= √13

Now, √13≠1.

Hence, from the above calculations, we can conclude that,

a is not a unit vector.

Formula To Find Unit Vector

â = a/|a|.......(2)

   = (x,y,z)/√(x2+y2+z2)

For example, a= (12,4,3)

Then, using (1),

|a|= √ ((122) +(42) +(32))

      = √144+16+9 = √169=13

Substituting, |a| in vector unit formula (2),

â = (12,4,3)/13

  = ((12/13), (4/13), (3/13))