AcademyVectors
Questions

Academy

Coordinate Bases

Level 1 - Math II (Physics) topic page in Vectors.

Principle

Cartesian coordinates describe a point or position vector by measuring signed distances along perpendicular axes. In three-dimensional space, the Cartesian basis vectors \(\mathbf i\), \(\mathbf j\), and \(\mathbf k\) point one unit along the \(x\)-axis, \(y\)-axis, and \(z\)-axis.

A basis is an ordered list of vectors that lets every vector in the space be written in exactly one way as a linear combination of those basis vectors. The same vector can have different components when a different basis is used.

Notation

\(\mathbf i\)
the unit vector in the positive x-direction of a Cartesian coordinate system
\(\mathbf j\)
the unit vector in the positive y-direction of a Cartesian coordinate system
\(\mathbf k\)
the unit vector in the positive z-direction of a Cartesian coordinate system
\(x\)
the signed component of a vector along \mathbf i
\(y\)
the signed component of a vector along \mathbf j
\(z\)
the signed component of a vector along \mathbf k
\(\mathbf p\)
a position vector or vector written with Cartesian components
\(D\)
the dimension of the vector space, meaning the number of vectors in a basis
\(\{\mathbf e_1,\ldots,\mathbf e_D\}\)
an ordered basis with D basis vectors
\(u_i\)
the coordinate scalar multiplying the basis vector \mathbf e_i
\(\mathbf u\)
a vector expressed in the ordered basis \{\mathbf e_1,\ldots,\mathbf e_D\}

The ordered basis \(\{\mathbf e_1,\ldots,\mathbf e_D\}\) has an order: \(\mathbf e_1\) is first, \(\mathbf e_2\) is second, and so on. Changing that order changes the meaning of a coordinate list.

Method

Step 1: Choose the coordinate system

Choose an origin \(O\), choose perpendicular axes, and choose one unit length on each axis. In a standard Cartesian system, \(\mathbf i\), \(\mathbf j\), and \(\mathbf k\) are unit vectors along those axes.

Step 2: Write the vector from its components

If \(\mathbf p\) has signed Cartesian components \(x\), \(y\), and \(z\), write

Cartesian decomposition
\[\mathbf p=x\mathbf i+y\mathbf j+z\mathbf k\]

The coordinate triple is written as

Coordinate triple
\[\mathbf p=(x,y,z)\]

Step 3: Compute magnitude from components

First apply Pythagoras in the \(xy\) plane. The projection of \(\mathbf p\) into the \(xy\) plane has legs \(x\) and \(y\), so its length \(r\) satisfies

Length in the xy plane
\[r^2=x^2+y^2\]
Plane length
\[r=\sqrt{x^2+y^2}\]

Then apply Pythagoras again, using the \(xy\)-plane length \(r\) and the vertical component \(z\):

Three-dimensional length
\[|\mathbf p|^2=r^2+z^2\]
Substitute the plane result
\[|\mathbf p|^2=(x^2+y^2)+z^2\]
Collect terms
\[|\mathbf p|^2=x^2+y^2+z^2\]
Take the non-negative square root
\[|\mathbf p|=\sqrt{x^2+y^2+z^2}\]

Step 4: Test whether proposed basis coefficients are unique

For a proposed ordered basis \(\{\mathbf e_1,\ldots,\mathbf e_D\}\), ask whether every vector \(\mathbf u\) can be written in one and only one way as

General basis expansion
\[\mathbf u=\sum_{i=1}^D u_i\mathbf e_i\]

In \(\mathbb R^2\), the ordered Cartesian basis \(\{(1,0),(0,1)\}\) gives unique coefficients. Suppose

Two expressions in the Cartesian basis
\[u_1(1,0)+u_2(0,1)=v_1(1,0)+v_2(0,1)\]

Compute each side component by component:

Left side
\[u_1(1,0)+u_2(0,1)=(u_1,0)+(0,u_2)=(u_1,u_2)\]
Right side
\[v_1(1,0)+v_2(0,1)=(v_1,0)+(0,v_2)=(v_1,v_2)\]
Equate components
\[(u_1,u_2)=(v_1,v_2)\]
First component
\[u_1=v_1\]
Second component
\[u_2=v_2\]

Since matching vector expressions force matching coefficients, the coefficients are unique.

For the ordered basis \(\{(1,0),(1,1)\}\), convert \((x,y)\) by solving

Non-standard basis equation
\[(x,y)=u_1(1,0)+u_2(1,1)\]

Work component by component:

Expand the right side
\[u_1(1,0)+u_2(1,1)=(u_1,0)+(u_2,u_2)\]
Add components
\[u_1(1,0)+u_2(1,1)=(u_1+u_2,u_2)\]
Equate to (x,y)
\[(x,y)=(u_1+u_2,u_2)\]
Second component
\[y=u_2\]
First component
\[x=u_1+u_2\]
Substitute u_2=y
\[x=u_1+y\]
Solve for u_1
\[u_1=x-y\]

Thus the coordinate scalars in this ordered basis are \(u_2=y\) and \(u_1=x-y\).

Rules

Coordinate triple
\[\mathbf p=(x,y,z)\]
Magnitude from Cartesian components
\[|\mathbf p|=\sqrt{x^2+y^2+z^2}\]
Cartesian basis expansion
\[\mathbf p=x\mathbf i+y\mathbf j+z\mathbf k\]
Basis expansion in dimension D
\[\mathbf u=\sum_{i=1}^D u_i\mathbf e_i\]

The coordinate scalars \(u_i\) are tied to the chosen ordered basis. If the basis changes, the numbers \(u_i\) can change even when the vector \(\mathbf u\) does not change.

Examples

Question
Find the magnitude of the vector
\[(0,3,-4)\]
Answer
Use
\[|\mathbf p|=\sqrt{x^2+y^2+z^2}\]
Here
\[x=0\]
\[y=3\]
and
\[z=-4\]
so
\[|\mathbf p|=\sqrt{0^2+3^2+(-4)^2}=\sqrt{0+9+16}=\sqrt{25}=5\]

Checks

  • Basis order matters: \((a,b)\) in \(\{\mathbf e_1,\mathbf e_2\}\) means \(a\mathbf e_1+b\mathbf e_2\), not \(a\mathbf e_2+b\mathbf e_1\).
  • Components depend on the basis, so the same vector can have different coordinate scalars in different bases.
  • A basis for \(\mathbb R^3\) needs three linearly independent vectors; fewer vectors cannot express every vector in \(\mathbb R^3\), and dependent vectors do not give unique coefficients.

Previous

Vector Spaces

Next

Dot Product