Unit Vector Calculator

An online unit vector calculator helps you to determine the components of any vector of length equal to 1 without changing the directions. Also, you can calculate the angle of a vector and the magnitude of an original vector with this normal vector calculator. Well, continue reading as we here tell you how to find the unit vector, the magnitude in simple steps, and a simple definition.

What is Unit Vector?

The vectors that have a magnitude equal to 1 are called unit vectors and it is donated by A^. Sometimes, it is also named as the multiplicative identity of a vector and direction vector. Usually, it is used for the direction of a vector. The length of the unit vector is one.

Unit Vector Formula:

A unit vector has one magnitude and they are donated with a “^” such as \(\hat{b}\). Moreover, a vector can be the unit vector after dividing it by the vector’s magnitude. Apart from this, they are often written in XY or XYZ coordinates. We can do it with two methods:

1. Write all coordinates in brackets: \( \vec {u} \) = (x, y, z).

2. Use three vectors XYZ, in which every point along the axes as: \( \vec {u} \) = \( x \hat {a} \) + \( y\hat {b} \) + \( z\hat {c} \).

Now, the magnitude of the vector is:

\(\mid \vec{u} \mid\) = \(\sqrt{x^{2} + y^{2} + z^{2}}\)

Unit vector = vector/ magnitude of the vector

However, an online normal unit vector calculator is helpful in finding the magnitude for the given values, its unit vector using its formula.

Representation of bracket format for unit vector calculation:

\( \hat {u} \) = \( \frac {\vec {u}} {\mid \vec {u} \mid} \) = \( \frac {(x, y, z)} {\sqrt {x^{2} + y^{2} + z^{2}}} \) = \( (\frac {x} {\sqrt {x^{^{2} + y^{2} + z^{2}}}}, \frac {y} {\sqrt {x^{2} + y^{2} + z^{2}}}, \frac {z} {\sqrt {x^{2} + y^{2} + z^{2}}}) \)

Representation of unit vector component format:

\(\hat{u}\) = \(\frac{\vec{u}}{\mid\vec{u} \mid}\) = \(\frac{(x\hat{i}, y\hat{j}, z\hat{k})}{\sqrt{x^{2}+y^{2}+z^{2}}} = \frac{x}{\sqrt{x^{^{2}+ y^{2}+ z^{2}}}}\hat{i}+ \frac{y}{\sqrt{x^{2}+ y^{2}+ z^{2}}}\hat{j}+ \frac{z}{\sqrt{x^{2}+ y^{2}+z^{2}}}\hat{k}\)

However, an online Unit Tangent Vector Calculator helps you to find the tangent vector of the vector value function at the given points.

Derivation of the unit vector formula:

\( \hat {v} \) = unit vector with one magnitude and direction

\( \vec {v} \) = vector with any direction and magnitude

\( \mid\vec {v} \mid\) = magnitude of the vector \( \vec{v} \)

x = vector in x axis

y = vector in y axis

z = vector in z axis

\( \hat{i} \) = unit vector bound for axis x

\( \hat{j} \) = unit vector bound for axis y

\( \hat{k} \) = unit vector bound for axis z

How to Find the Unit Vector?

For instant calculations, you just have to add values of components into the magnitude and direction calculator and let it do it all in seconds. And even you can perform manual calculations with these simple steps:

Example:

Find out the unit vector of a vector u = (8, -3, 5) in the same direction.

Solution:

First, write all components of the vector as:

$$ a_1 = 8, b_1 = -3, c_1 = 5 $$

Now, calculate the magnitude of u vector:

$$ |u| = \sqrt{(a_1^2 + b_1^2 + c_1^2)} $$

$$ |u| = \sqrt{(8^2 + (-3)^2 + 5^2)} $$

$$ |u| = \sqrt{(64 + 9 + 25)} $$

$$ |u| = \sqrt{(98)} $$

$$ |u| = 9.9 $$

In case, if you are worried about the magnitude of any vector, then use our handy unit vector calculator for determining the magnitude precisely. So, the magnitude of vector u is 9.9. Now, divide each vector component by | u |.

a_2 = a_1 / | u | = 8 / 9.9 = 0.8081

b_2 = b_1 / | u | = -3 / 9.9 = -0.3031

c_2 = c_1 / | u | = 5 / 9.9 = 0.5051

Now, write all the obtained values in vector form as:

$$ u = (0.8081, -0.3031, 0.5051) $$

Lastly, take the sum of all these values. If the magnitude is 1, then the result is correct. However, an online Directional Derivative Calculator determines the directional derivative and gradient of a function at a given point of a vector.

Example 1:

Calculate the unit vector of a vector v = {3, 4},in two-dimensional space.

Solution:

Given Values:

x = 3

y = 4

The magnitude of vector:

$$ \vec{v} = 5 $$

The vector direction calculator finds the direction by using the values of x and y coordinates. So, the direction Angle θ is: $$ θ = 53.1301 deg $$

The unit vector is calculated by dividing each vector coordinate by the magnitude. So, the unit vector is: $$ \(vec{e} = (3/5, 4/5 \) $$ $$ \(vec{e} = (0.6, 0.8)\ $$

Example 2:

Find the missing unit vector component z in three-dimensional space, where x is 0.9 and y is 0.4.

Solution:

Given Values:

$$ x = 0.9 $$

$$ y = 0.4 $$

Unit vector:

$$ (x, y, z) = (0.9, 0.4, 0.137) $$

$$ Magnitude = 1 $$

Spherical Coordinate Unit Vector:

Well, the function position is the unit vector in the spherical system. To find the spherical coordinate unit in the form of rectangular coordinate systems that are not the function of position.

r^ = r^r = xx^+yy^+zz^r = x^sinθ cosØ + y^sinθ sinØ - z^cosθ

z×r^Sinθ = - x^sinØ + y^sinØ

How does Unit Vector Calculator Works?

An online unit normal vector calculator determines the unit vector in the direction of entered vectors by following these steps:

Input:

• First, select a method and dimensions of a vector from the drop-down list.
• When you choose 2D or 3D dimensions from the list, the calculator shows different diagrams for both options.
• Then, substitute all corresponding values in the fields.
• Click on the “Calculate Unit Vector” button.

Output:

• The normal vector calculator computes the magnitude of a vector, angle direction, and unit vector.

FAQ:

Does the unit vector have any units?

No, a unit vector has not any unit or dimensions, it has only directions.

What is a Polar Vector?

The vector that has either an application point or a starting point is defined as a polar vector. Velocity is the best example of a polar vector.

What is the magnitude of the zero vector?

A zero vector is the null vector with zero magnitudes. The velocity of stationary objects is an example of a zero vector.

When are two vectors equal?

Two vectors are equal if they represent the same magnitude and direction.

Are unit vectors always perpendicular?

No, the unit vectors are not perpendicular all the time. These vectors are always tangent and parallel to the original vector.

Conclusion:

An online unit vector calculator computes the unit vector, magnitude, and angle of the vector from another vector. This online calculator is more beneficial for engineering and science students, which are used to solve physics and mathematical problems in their daily life routine.

Reference:

From the source of Wikipedia: Cartesian coordinates, cylindrical coordinates, Spherical coordinates, General unit vectors, Curvilinear coordinates, Right versor. From the source of Topper: Unit Vector Formula, Derivation of the unit vector formula, Euclidean space, cross product of two random vectors. From the source of Vedantu: What is unit vector in physics, formula, Spherical Coordinate, Unit Tangent Vector, Unit Normal Vector, Orthogonal Unit Vector, Vector Perpendicular to Two Vectors.