When you need to form the basis of the vector space, you need to work on the unit vector. Basically, each vector that exists in the space has a linear combination of vectors that we call unit vectors. So, when it comes to the definition of the unit vector, the dot product of two vectors in a Euclidean space is a scalar value that amounts to the cosine of the subtended angle. This angle is typically smaller.
A unit vector basically has a length “1.” To obtain this vector in the direction of any other vector, you need to divide the vector “A” by its module. So, mathematically, you will represent it in the form of ˆa. aˆ = a |a|.
Also, every non-zero vector has a unit vector. We, sometimes, call it the corresponding vector. It is because this vector has the same direction and has a magnitude of “1.” So, if you want to find the unit vector “U” of any vector, then you need to focus on:
V = <X, Y>
Then, you need to divide the vector by taking its magnitude. Mathematically, you will do it the following way.
U = V / |v|
Keep in mind that the formula given above uses scalar multiplication. It is because the denominator is a scalar and the numerator is a vector. A scalar represents the real number. A scalar usually scales the vector. Simply put, it changes the value or scale of the vector.
Let us give you an example. The real number “3” scales the vector “V” by a factor of “3”. So, the 3V is thrice as long as “V.”
How to find the unit vector?
Today, we find many online calculators, which have been programmed to find the unit vector. The calculators are even helpful in finding the unit vector step-by-step. However, to learn the rules, you must find it manually. This is the rule of mathematics that we prefer and encourage others. This way, you can improve your calculation skills.
For example, if you want to find the unit vector of the vector “V” in the same direction with its components are V (vector) = (2 – 1) V “vector = (2, -1).
So, the first step is to use the formula for calculating the magnitude of the given vector. Next, you need to multiply the vector by the reciprocal of its magnitude. Make sure the reciprocal is negative. Also, the calculation is obtaining the unit vector in the opposite direction.
You can transform the vector into another vector that has a length value of “1” without changing the direction. You must obtain the result of the unit vector by dividing all the elements of any vector by its magnitude.
What is the formula of Unit Vector?
You can calculate the unit vector of any arbitrary vector in the same diction. To find the unit vector, you need to apply this formula.
û = u / |u|
In this formula, the “û” is the unit vector. Also, we represent the arbitrary vector by “u,” which is in the form of (x, y, and z). The module of the vector u that is |u| represents the magnitude of the arbitrary vector. So, to calculate the vector’s magnitude, you need to use the formula or the equation:
|u| = √(x² + y² + z²)
Using this equation, it is easy to measure the magnitude of the vector “u.” The simple way is to find the midpoint of the segment.
Unit Vector Example
For example, we want to calculate the unit vector the values of which are (8, -2, and 5). If you want the calculation for the same direction, you need to write down the components of the vector such as x, y, and z. In this example, we have x1 = 8, y1 = -2 and z1 = 5. You can calculate the magnitude of the vector in the following way.
|u| = √ (x₁² + y₁² + z₁²)
|u| = √ (8² + (-2)² + 5²)
|u| = √ (64 + 6 + 25)
|u| = √95
|u| = 9.74
After finding the magnitude, the next step is to find the unit vector. Let us do this for the same example in the next step.
Once you have the magnitude of the “u” vector, you can easily find the unit vector. You need to divide each of the components of the vector by the module of the vector – i.e. |u|.
X₂ = 8 / 9.74 = 0.821
Y₂ = -2 / 9.74 = — 0.20
Z₂ = 5 / 9.74 = 0.51
So, you have got the results, which are 0.821, -0.20, and 0.51. The last step is to check the results of the unit vector and determine whether or not they are correct. Make sure the magnitude of the vector “u” is equal to 1. This way, the results are accurate.
What is the unit of unit vector?
How do you find the unit vector of a line?
How do you find the unit vector given two points?
How do you tell if a vector is a unit vector?
What is a vector formula?
What is the formula for resultant vector?
The rules for finding the resultant of a vector or adding more than two vectors can be protracted to any number of vectors. R=A+B+C+………………………….