Find The Coordinate Vector Calculator

Find the Coordinate Vector

Enter the components of the vector v and the basis vectors b1 and b2 (for a 2D space) to find the coordinate vector of v with respect to the basis {b1, b2}.

Vector	Component 1	Component 2
v	5	4
b1	2	1
b2	1	3

Input vectors used in the calculation.

What is a Coordinate Vector?

A coordinate vector represents a vector from a vector space as a unique linear combination of basis vectors from a chosen basis. If we have a vector space V and a basis B = {b1, b2, …, bn} for V, then any vector v in V can be uniquely written as v = c1*b1 + c2*b2 + … + cn*bn, where c1, c2, …, cn are scalars. The coordinate vector of v with respect to the basis B is then the column vector [c1, c2, …, cn]^T, often denoted as [v]_B.

Essentially, the coordinate vector tells us “how much” of each basis vector we need to combine to form the original vector v. The components of the coordinate vector are the coefficients of the linear combination.

This concept is fundamental in linear algebra and is used when we want to represent vectors in different coordinate systems defined by different bases. Anyone studying or working with linear algebra, physics, engineering, computer graphics, or data science will likely use the concept of a coordinate vector.

A common misconception is that a vector has only one set of coordinates. However, a vector’s coordinates change depending on the basis used to describe it. The standard basis (e.g., {[1,0], [0,1]} in 2D) gives the most familiar coordinates, but any set of linearly independent vectors spanning the space can form a basis, leading to a different coordinate vector for the same vector v.

Coordinate Vector Formula and Mathematical Explanation (2D Case)

Let’s consider a 2-dimensional vector space with a basis B = {b1, b2}, where b1 = [b11, b12]^T and b2 = [b21, b22]^T. Let v = [v1, v2]^T be a vector in this space. We want to find the coordinate vector [v]_B = [c1, c2]^T such that:

v = c1*b1 + c2*b2

This vector equation translates into a system of linear equations:

v1 = c1*b11 + c2*b21

v2 = c1*b12 + c2*b22

We can write this in matrix form as: A*c = v, where:

A = [[b11, b21], [b12, b22]] (matrix whose columns are the basis vectors)

c = [c1, c2]^T (the coordinate vector we want to find)

v = [v1, v2]^T

To solve for c, we can use the inverse of matrix A, provided A is invertible (i.e., its determinant is non-zero, meaning b1 and b2 are linearly independent):

c = A^(-1)*v

The determinant of A is det(A) = b11*b22 – b21*b12.

If det(A) ≠ 0, the inverse A^(-1) is: (1/det(A)) * [[b22, -b21], [-b12, b11]]

So, the coordinates c1 and c2 are:

c1 = (1/det(A)) * (b22*v1 – b21*v2)

c2 = (1/det(A)) * (-b12*v1 + b11*v2)

The coordinate vector of v with respect to basis B is [v]_B = [c1, c2]^T.

Variables Table

Variable	Meaning	Unit	Typical Range
v1, v2	Components of vector v	Depends on context (e.g., meters, m/s)	Real numbers
b11, b12	Components of basis vector b1	Same as v	Real numbers
b21, b22	Components of basis vector b2	Same as v	Real numbers
c1, c2	Coordinates of v relative to B	Scalar (dimensionless)	Real numbers
det(A)	Determinant of the basis matrix	Depends on context	Real number

Practical Examples (Real-World Use Cases)

Example 1: Standard vs. Rotated Basis

Suppose we have a vector v = [3, 4]^T in the standard basis. Let’s find its coordinates with respect to a new basis B = {b1, b2} where b1 = [1, 1]^T and b2 = [-1, 1]^T (a rotated and scaled basis).

Here, v1=3, v2=4, b11=1, b12=1, b21=-1, b22=1.

det(A) = (1)(1) – (-1)(1) = 1 + 1 = 2.

c1 = (1/2) * ((1)(3) – (-1)(4)) = (1/2) * (3 + 4) = 3.5

c2 = (1/2) * ((-1)(3) + (1)(4)) = (1/2) * (-3 + 4) = 0.5

So, the coordinate vector of v with respect to B is [v]_B = [3.5, 0.5]^T. This means v = 3.5*b1 + 0.5*b2.

Example 2: Skewed Basis

Let v = [5, 4]^T, and the basis be B = {b1, b2} with b1 = [2, 1]^T and b2 = [1, 3]^T (as in the calculator’s default).

v1=5, v2=4, b11=2, b12=1, b21=1, b22=3.

det(A) = (2)(3) – (1)(1) = 6 – 1 = 5.

c1 = (1/5) * ((3)(5) – (1)(4)) = (1/5) * (15 – 4) = 11/5 = 2.2

c2 = (1/5) * ((-1)(5) + (2)(4)) = (1/5) * (-5 + 8) = 3/5 = 0.6

The coordinate vector is [v]_B = [2.2, 0.6]^T. So, v = 2.2*[2, 1]^T + 0.6*[1, 3]^T = [4.4, 2.2]^T + [0.6, 1.8]^T = [5, 4]^T, which matches our original v.

How to Use This Coordinate Vector Calculator

1. Enter Vector v: Input the components v1 and v2 of the vector v for which you want to find the coordinates.
2. Enter Basis Vector b1: Input the components b11 and b12 of the first basis vector.
3. Enter Basis Vector b2: Input the components b21 and b22 of the second basis vector.
4. Calculate: The calculator automatically updates the results as you type. You can also click “Calculate”.
5. View Results:
   • Primary Result: Shows the coordinate vector [c1, c2].
   • Intermediate Values: Displays the determinant of the basis matrix, and the individual values of c1 and c2.
   • Formula Explanation: A brief reminder of the formula used.
   • Table: Summarizes the input vectors.
   • Chart: Visualizes v, b1, b2, and the components c1*b1 and c2*b2 summing to v.
6. Reset: Click “Reset” to return to default values.
7. Copy Results: Click “Copy Results” to copy the main result, intermediate values, and formula to your clipboard.

If the determinant is zero, the basis vectors are linearly dependent, and a unique coordinate vector may not exist for the given vector v, or there might be infinite solutions if v is in the span of b1 and b2. The calculator will indicate this.

Key Factors That Affect Coordinate Vector Results

1. Choice of Basis Vectors (b1, b2): The coordinate vector is entirely dependent on the basis chosen. Different bases yield different coordinates for the same vector v.
2. Linear Independence of Basis Vectors: The basis vectors must be linearly independent (determinant of the basis matrix must be non-zero) for a unique coordinate vector to exist for any v in the space. If they are dependent, they don’t span the entire space (e.g., in 2D, they lie on the same line), or they are redundant.
3. Components of Vector v: The coordinates c1 and c2 directly depend on the values of v1 and v2.
4. Order of Basis Vectors: Swapping b1 and b2 will swap c1 and c2 in the resulting coordinate vector and change the basis matrix.
5. Scale of Basis Vectors: If you scale the basis vectors, the coordinates will scale inversely to keep the linear combination equal to v.
6. Dimensionality: This calculator is for 2D. In 3D or higher dimensions, you’d have more basis vectors and more components for each vector, leading to a larger system of equations and a larger coordinate vector.

Frequently Asked Questions (FAQ)

What is a basis?

A basis for a vector space is a set of linearly independent vectors that span the entire space. This means any vector in the space can be written as a unique linear combination of the basis vectors.

What if the determinant is zero?

If the determinant of the basis matrix is zero, the basis vectors are linearly dependent. They do not form a valid basis for the 2D space (they might span only a line). In this case, either no solution exists for c1, c2, or infinitely many solutions exist if v happens to be in the span of b1 and b2. Our calculator will indicate this.

Can I use this for 3D vectors?

No, this specific calculator is designed for 2D vectors and a 2-vector basis. For 3D, you would need three basis vectors and solve a 3×3 system of equations. See our matrix calculator for solving 3×3 systems.

What is the standard basis in 2D?

The standard basis in 2D is usually {e1, e2} where e1 = [1, 0]^T and e2 = [0, 1]^T. The coordinates of a vector with respect to the standard basis are just its own components.

Why are coordinate vectors useful?

They allow us to represent vectors in different coordinate systems, which can simplify problems. For example, changing to a basis of eigenvectors can diagonalize a matrix, simplifying calculations involving linear transformations. They are crucial in change of basis operations.

Is the coordinate vector unique?

Yes, for a given vector and a given basis, the coordinate vector is unique.

What does it mean for vectors to be linearly independent?

A set of vectors is linearly independent if none of them can be written as a linear combination of the others. In 2D, two vectors are linearly independent if they are not parallel (or one is not the zero vector).

How is the coordinate vector related to a linear combination?

The components of the coordinate vector are the coefficients (scalars) used in the linear combination of the basis vectors to form the original vector.

Related Tools and Internal Resources

• Linear Algebra Basics: Learn the fundamentals of vectors, matrices, and vector spaces.
• Matrix Calculator: Perform various matrix operations, including finding determinants and inverses, and solving systems of linear equations.
• Vector Addition Calculator: Add or subtract vectors.
• Dot Product Calculator: Calculate the dot product of two vectors.
• Eigenvalue and Eigenvector Calculator: Find eigenvalues and eigenvectors of a matrix, often used in changing to a basis of eigenvectors.
• Basis and Dimension: Understand the concepts of basis and dimension of a vector space more deeply.