2.4: Lines and planes through the origin in ℝ³

In Chapter 2.1, we learned how to describe a line in $\mathbb R^2$ using scalar multiples of a nonzero vector. We’ll use the same idea here, with one extra coordinate, and then describe planes using linear combinations of two vectors.

This section, along with the rest of Chapter 2, contains several 3D figures. Drag them with your mouse to view the lines and planes from different perspectives.

Here’s a video introducing what a vector looks like in $\mathbb{R}^3$.

Lines through the origin¶

Remember that the span of one vector is the set of all its scalar multiples. Let’s start with

Multiplying $\vec v$ by a scalar changes its length and possibly reverses its direction. If we draw all of these multiples from the origin, their tips trace out a line! As before, we can write this line as

For instance, $t=1$ gives $(3,4,5)$, $t=-1$ gives $(-3,-4,-5)$, and $t=0$ gives the origin. The parameter can be any real number; the figure only shows part of the line.

To move from vector form to scalar form, read off the entries. To move back, collect the coefficients of the parameter into a vector.

Here’s a video showing how to visualize a line in $\mathbb{R}^3$ using Desmos 3D.

Here’s a video that visualizes this idea further.

Planes through the origin¶

Linear independence¶

To fill a plane through the origin, we need two directions that cannot be obtained by scaling one another. Let’s give this condition a name.

We’ll often shorten “linearly independent” to just “independent.”

For two nonzero vectors, independence means that they point along different lines. For example, $\begin{bmatrix}1\\0\\0\end{bmatrix}$ and $\begin{bmatrix}0\\1\\0\end{bmatrix}$ are independent and span the $xy$-plane. But $\begin{bmatrix}1\\0\\0\end{bmatrix}$ and $\begin{bmatrix}2\\0\\0\end{bmatrix}$ are dependent and span only the $x$-axis. The second vector adds no new direction.

This definition of linear independence works for a pair of vectors. In later chapters, we’ll discuss what it means for three or more vectors to be linearly independent.

Span of two vectors¶

Back to the main idea: describing a plane in $\mathbb{R}^3$.

For example, let

These vectors are independent: matching the second entry of $\vec v_1$ would require multiplying $\vec v_2$ by 2, but that would give a first entry of 10, not 3. So, there’s no number we can multiply $\vec v_2$ by to get $\vec v_1$.

Notice that we’re asking for less than we did in Chapter 2.2. Our vectors don’t need to have length one, and they don’t need to be perpendicular. We just need two independent directions in the plane.

For example, consider $2\vec v_1-3\vec v_2$, a linear combination of $\vec v_1$ and $\vec v_2$:

Draw $2\vec v_1$ from the origin, then draw $-3\vec v_2$ from the tip of $2\vec v_1$. The vector from the origin to the final tip is $2\vec v_1-3\vec v_2$.

Notice that $2\vec v_1-3\vec v_2$ also lives on this same plane. In fact, every linear combination of $\vec v_1$ and $\vec v_2$ lives on this plane, and every vector that lives on this plane can be written as a linear combination of $\vec v_1$ and $\vec v_2$!

In our example, this gives

You can think of $a$ and $b$ as two knobs we can turn: $a$ tells us how much of $\vec v_1$ to use, and $b$ tells us how much of $\vec v_2$ to use. Letting both range over all real numbers gives the entire plane.

Parametric equations¶

The same representation terminology applies to planes. Both of the following are vector-parametric forms of $P$:

Expanding the vector sum gives

Reading the entries gives the scalar-parametric form:

In the scalar-parametric form above, all three equations use the same two parameters. Each pair $(a,b)$ selects one point; allowing both parameters to range independently over all real numbers fills the plane.

Note that the letters $a$ and $b$ are arbitrary. We could, and often do, use $s$ and $t$ as well.

We can easily move back and forth between the two parametric descriptions of a plane. For example, suppose we’re given the scalar-parametric form

Collecting the coefficients of $s$ and $t$ gives

Equivalently, this plane is

or even

Here’s a video showing how to visualize a plane in $\mathbb{R}^3$ using Desmos 3D.

Different vectors spanning the same plane¶

Recall, the plane $P$ is defined as the span of $\vec v_1$ and $\vec v_2$,

Key idea: the exact same plane, $P$, can also be written as the span of other pairs of vectors! To illustrate, let’s define two new vectors, $\vec v_3$ and $\vec v_4$, using our original two:

The resulting vectors

also span $P$:

Why? Each pair can be built from the other pair. That means anything we can build using one pair can also be built using the other. Let’s check this carefully.

• By construction, any linear combination of $\vec v_3$ and $\vec v_4$ is also a linear combination of $\vec v_1$ and $\vec v_2$:

  $$\begin{aligned} c\vec v_3+d\vec v_4 &=c(\vec v_1-\vec v_2)+d(2\vec v_1+3\vec v_2)\\ &=(c+2d)\vec v_1+(-c+3d)\vec v_2. \end{aligned}$$

  So every vector in $\operatorname{span}(\vec v_3,\vec v_4)$ belongs to $P$.

• But that’s only half of what we need. We also need to show that we can build $\vec v_1$ and $\vec v_2$ using $\vec v_3$ and $\vec v_4$. Notice that

  $$\begin{aligned} 3\vec v_3+\vec v_4 &=3(\vec v_1-\vec v_2)+(2\vec v_1+3\vec v_2)=5\vec v_1,\\ \vec v_4-2\vec v_3 &=(2\vec v_1+3\vec v_2)-2(\vec v_1-\vec v_2)=5\vec v_2 \end{aligned}$$

  so

  $$\vec v_1=\frac35\vec v_3+\frac15\vec v_4,\qquad \vec v_2=-\frac25\vec v_3+\frac15\vec v_4.$$

  Therefore, any vector in $P$ can also be written as

  $$\begin{aligned} a\vec v_1+b\vec v_2 &=a\left(\frac35\vec v_3+\frac15\vec v_4\right) +b\left(-\frac25\vec v_3+\frac15\vec v_4\right)\\ &=\frac{3a-2b}{5}\vec v_3+\frac{a+b}{5}\vec v_4. \end{aligned}$$

So, we can move back and forth between the two descriptions: any linear combination of $\vec v_1$ and $\vec v_2$ is also a linear combination of $\vec v_3$ and $\vec v_4$, and vice versa. The coefficients change, but the set of vectors we can reach stays the same!

More generally, any two linearly independent vectors in $P$ span $P$. As in Chapter 2.1, a span description is not unique.

For our original plane, the replacement vectors above give another vector-parametric form:

Its scalar-parametric form is

The point $(3,4,5)$ occurred at $a=1,b=0$ in the original parametrization. Here it occurs at $s=3/5,t=1/5$. The same point can have different parameter values in different parametrizations.

Here’s a video synthesizing the main ideas of this section.

What’s next? This section described how to express lines and planes in $\mathbb{R}^3$ in parametric form, using the fact that they can be thought of as spans of vectors. In Chapter 2.5, we will describe lines and planes in $\mathbb{R}^3$ using linear equations: that is, equations of the form