Unit Tangent Vector Calculator

An online unit tangent vector calculator helps determine the tangent vector of a vector-valued function at specific points. It also shows the derivative of trigonometric functions and provides the normalized form. Read on to learn the unit tangent vector formula and how to compute tangent vectors with examples.

What is a Unit Tangent Vector?

In mathematics, the Unit Tangent Vector is the derivative of a vector-valued function, which provides another vector-valued function that is unit tangent to the defined curve. The direction of the tangent line is similar to the slope of the tangent line. Since the vector contains magnitude and direction, the velocity vector contains more information than we need. We can strip its magnitude by dividing its magnitude.

Unit Tangent Vector Formula:

Let r(t) be a function with differentiable vector values, and v(t) = r’(t) be the velocity vector. Then, the tangent vector equation is the unit vector in the direction of the velocity vector which is used by the unit tangent vector calculator to find the length of the vector.

$$T(t) = \frac{v(t)}{||v(t)||}$$

However, an Online Derivative Calculator allows you to find the derivative of the function with respect to a given variable.

Example:

Finding unit tangent vectorT(t) and T(0).

Let

$$r(t) = t a + e^t b - 2t^2 c$$

Solution:

We have

$$v(t) = r'(t) = a + e^t b - 4t c$$

and

$$||v(t)|| = \sqrt{1 + e^{2t} + 16 t^2}$$

Unit tangent vector:

$$T(t) = \frac{v(t)}{||v(t)||} = \frac{a + e^t b - 4t c}{\sqrt{1 + e^{2t} + 16 t^2}}$$

At t = 0:

$$ \begin{aligned} T(0) &= a + e^0 b - 4(0)c / \sqrt{1 + e^{2(0)} + 16(0)^2} \\ \end{aligned} $$

$$ = a + b / \sqrt{2} $$

$$ = 1 / \sqrt{2}a + 1 / \sqrt{2}b $$

What is the Principle of Unit Normal Vector?

The normal vector is the perpendicular vector. For a vector v in space, there are infinitely several perpendicular vectors. Our aim is to choose a special vector that is perpendicular to the unit tangent vector. For non-straight curves, this vector is geometrically the only vector pointing to the curve. Algebraically, we can use the following definitions to calculate vectors.

Let r(t) be a differentiable vector function, and let T(t) be a tangent vector. Then the normal vector N(t) of the principle unit is defined as

$$N(t) = \frac{T'(t)}{||T'(t)||}$$

This equation is used by the unit tangent vector calculator to find the norm (length) of the vector. If it is compared with the tangent vector equation, then it is regarded as a function with vector value. The principle unit normal vector is the tangent vector of the vector function.

However, an Online Instantaneous Velocity Calculator allows you to calculate instantaneous velocity corresponding to the instantaneous rate of change of velocity formula.

Find Normal and Tangential Components of Acceleration:

When driving, you will encounter two forces, which will change your velocity. The car accelerates under the action of gravity. The second change in speed is caused by the car turning. The first component of acceleration is called the tangential component of acceleration, and the other component is the normal component of acceleration. It is assumed that the tangential component of acceleration is along the direction of the vector of the tangent unit, and the normal component of acceleration is along the direction of the normal vector of the principle unit. When we have T and N, it is easy to find two components.

The tangential component of acceleration is

$$ a_T = a \cdot T = \frac{v \cdot a}{||v||} $$

and the normal component of acceleration is

$$ a_N = a \cdot N = \frac{||v \times a||}{||v||} $$

and

$$a = a_N N + a_T T$$

How Unit tangent Vector Calculator Works?

The tangent vector calculator determines the unit tangent vector of a function at a point by follow these instructions:

Input:

Firstly, enter a function with different trigonometric values such as sine, cosine, and tangent.
Now, enter a point to find the unit tangent vector.
To see the results, hit the "CALCULATE" button.

Output:

The tangential velocity calculator provides input and answers at a given point.
This calculator displays a single derivation for every trigonometric function and normalize form, also find the unit tangent vector with stepwise calculations.

FAQ:

Binormal vector a unit vector. How?

Since the binormal vector is defined as the cross product of the unit tangent vector and the unit normal vector, also it is orthogonal to both the normal vector and the tangent vector.

How to find the tangential?

Divide the circumference by the time it takes to find the tangential speed for completing one revolution.

Which units are used for tangential velocity?

The tangent velocity formula is used to calculate the tangential velocity of objects in a circular motion. Expressed in meters per second (m/s).

What is the difference between tangential velocity and angular velocity?

Angular velocity is the rate at which an angle (radians) changes over time, expressed in units of 1/s. While tangential velocity is the velocity of a point on the surface of a rotating object that is multiplied by the distance from the point of the axis of rotation.

Conclusion:

Use this online unit tangent vector calculator for finding the normalized form and the tangential vector of a function. Also, this calculator differentiates the function and computes the length of a vector at given points.

References:

Wikipedia: Tangent Vector, Contravariance, Tangent vectors on manifolds.

Ximera: Unit Tangent & Normal Vectors, Normal Components of Acceleration.

Oregon State: Derivative of a Vector Function, Unit Tangent Vector, Arc Length