04 Planes and linear spaces, part 2

Exercise 1: Temperatures

Suppose a \(3\)-dimensional coordinate system has been set up in the classroom. The temperature, \(T\), in degrees Fahrenheit, at a point in the classroom is a function of the coordinates \((x,y,z)\) of that point, \(T = f(x,y,z)\). Suppose that distances are measured in feet, and that

\[ T = f(x,y,z) = 0.5x + 0.25y - 0.1z + 70. \]

What is the temperature at the point \((0,0,0)\)?
What is the temperature at the point \((2,4,10)\)?
For every foot moved in the positive \(x\)-direction, how much does the temperature change?
For every foot moved in the positive \(y\)-direction, how much does the temperature change?
For every foot moved in the positive \(z\)-direction, how much does the temperature change?
For every foot moved in the negative \(z\)-direction, how much does the temperature change?

Exercise 2: Production costs

A factory produces custom furniture. The total cost to produce one item of furniture, \(C\), in dollars, depends on four variables:

\(x\) = number of workers assigned
\(y\) = hours of machine time used
\(z\) = square feet of wood used
\(w\) = gallons of finish/stain applied

The cost function is: \[ C = f(x,y,z,w) = 25x + 15y + 3z + 12w + 200. \]

What is the base cost (when \(x=y=z=w=0\))?
What is the cost when \(x=4\) workers, \(y=8\) hours of machine time, \(z=50\) square feet of wood, and \(w=2\) gallons of finish are used?
How much does the cost increase for each additional worker?
How much does the cost increase for each additional square foot of wood?
How much does the cost increase for each additional gallon of finish?

What do these exercises have in common?

The function in Exercise 1 is of the form \[ w = m(x-x_0) + n(y-y_0) + p(z-z_0) + w_0. \]
The function in Exercise 2 is of the form \[ u = m(x-x_0) + n(y-y_0) + p(z-z_0) + q(w-w_0) + u_0. \]
Do you see the pattern?!

Point-slope equations of “hyperplanes”

In two dimensions, we use the coordinates \((x,y)\).
In three dimensions, we use the coordinates \((x,y,z)\).
In four dimensions, we use the coordinates \((x,y,z,w)\).
In five dimensions, we use the coordinates \((x,y,z,w, u)\).
In \(n\) dimensions, we use the coordinates \((x_1, x_2, \ldots, x_n)\).

Point-slope equation for a hyperplane in \(\mathbb{R}^4\)

A point-slope equation for a hyperplane through a point \((x_0,y_0, z_0, w_0)\) with

slope \(m\) in the positive \(x\)-direction,
slope \(n\) in the positive \(y\)-direction, and
slope \(p\) in the positive \(z\)-direction

is \[ w = m(x-x_0) + n(y-y_0) + p(z-z_0) + w_0. \]

Up one more dimension:

Point-slope equation for a hyperplane in \(\mathbb{R}^5\)

A point-slope equation for a hyperplane through a point \((x_0,y_0, z_0, w_0, u_0)\) with

slope \(m\) in the positive \(x\)-direction,
slope \(n\) in the positive \(y\)-direction,
slope \(p\) in the positive \(z\)-direction, and
slope \(q\) in the positive \(w\)-direction

is \[ u = m(x-x_0) + n(y-y_0) + p(z-z_0) + q(w-w_0) + u_0. \]

So, the graphs of the functions in Exercise 1 and 2 were hyperplanes in \(\mathbb{R}^4\) and \(\mathbb{R}^5\), respectively.

The full story

Point-slope equation for a hyperplane in \(\mathbb{R}^n\)

A point-slope equation for a hyperplane in \(\mathbb{R}^n\) is \[ x_n = m_1(x_1-p_1) + m_2(x_2-p_2) + \cdots + m_{n-1}(x_{n-1}-p_{n-1}) + p_{n}, \]

where \((p_1, p_2, \ldots, p_{n})\) is a point in \(\mathbb{R}^n\) and \(m_1, m_2, \ldots, m_{n-1}\) are the slopes in the positive \(x_1\), \(x_2\), \(\ldots\), \(x_{n-1}\) directions.

You can’t see the graphs of these hyperplanes when \(n>3\), but that doesn’t mean these are imaginary objects with no real-world applications.
If you doubt this, just look back at Exercises 1 and 2.

Vector, standard and affine equations

Vector equation for a hyperplane in \(\mathbb{R}^4\)

A vector equation for a hyperplane in \(\mathbb{R}^4\) is \[ \langle \mathbf{n}, \mathbf{r}-\mathbf{r}_0 \rangle = 0, \]

where \(\mathbf{n}\) is a normal vector to the hyperplane, \(\mathbf{r}_0\) is the position vector of a point on the hyperplane, and \(\mathbf{r}\) is a variable position vector.

Standard equation for a hyperplane in \(\mathbb{R}^4\)

A standard equation for a hyperplane in \(\mathbb{R}^4\) is \[ a(x-x_0) + b(y-y_0) + c(z-z_0) + d(w-w_0) = 0, \]

where \((x_0,y_0,z_0,w_0)\) is a point on the hyperplane and \(a,b,c,d\) are the components of a normal vector to the hyperplane.

Affine equation for a hyperplane in \(\mathbb{R}^4\)

An affine equation for a hyperplane in \(\mathbb{R}^4\) is \[ ax + by + cz + dw = e, \]

where:

\(a\), \(b\), \(c\), \(d\), and \(e\) are constants,
not all of \(a\), \(b\), \(c\), and \(d\) are \(0\), and
the numbers \(a\), \(b\), \(c\), and \(d\) are the components of a normal vector to the hyperplane.

Can you guess what these equations would look like in \(\mathbb{R}^5\)? In \(\mathbb{R}^n\)?

Exercise 3: Practice with equations of hyperplanes

a. Point-slope equation to standard equation

Consider the hyperplane in \(\mathbb{R}^4\) with point-slope equation \[ w = 2(x-1) - (y-2) + 3(z-3) + 4. \]

Identify a point on the hyperplane. Write the equation in standard form and identify a normal vector to the hyperplane.

b. Affine equation to standard equation

Consider the hyperplane in \(\mathbb{R}^4\) with affine equation \[ 2x - y + 3z - 3w = 4. \]
Identify a normal vector to the hyperplane. Write the equation in standard form and identify a point on the hyperplane.

Exercise 4: Identifying points on hyperplanes

Consider the hyperplane in \(\mathbb{R}^5\) with affine equation

\[ 3x - 2y + z + 4w - 5u = 7. \]

Identify a normal vector to the hyperplane.
Identify a point on the hyperplane.
Where does the hyperplane intersect the \(x\)-axis?
Where does the hyperplane intersect the \(y\)-axis?

Exercise 5: Level sets

a. Level sets of linear functions of two variables

Consider the linear function with point-slope equation \[ z = f(x,y) = 2x - 2y + 3. \]

If we intersect the graph of \(f\) with the plane \(z = 0\), what sort of shape do we get?
If we intersect it with the plane \(z = 1\), what sort of shape do we get?
What about the planes \(z = -1\) and \(z = 2\)?

These intersections are called level sets of the function.

b. Level sets of linear functions of three variables

Consider the linear function with point-slope equation

\[ w = f(x,y,z) = 2x - y + 3z + 4. \]

If we intersect the graph of \(f\) with the hyperplane \(w = 0\), what sort of shape do we get?
If we intersect it with the hyperplane \(w = 1\), what sort of shape do we get?

These intersections are called level sets of the function.