DLA Discovery guide

Discovery guide 14.1 Discovery guide

Recall that for $\uvec{u} = (u_1,u_2,\dotsc,u_n)$ and $\uvec{v} = (v_1,v_2,\dotsc,v_n)\text{,}$ the dot product of $\uvec{u}$ and $\uvec{v}$ is defined by the formula given below on the left. It is an important formula because if $\theta$ is the angle between two nonzero vectors $\uvec{u}$ and $\uvec{v}\text{,}$ then $\theta$ satisfies both $0 \le \theta \le \pi$ and the formula given below on the right.

\begin{align*} \udotprod{u}{v} \amp= u_1 v_1 + u_2 v_2 + \dotsb + u_n v_n \amp \cos\theta \amp= \frac{\udotprod{u}{v}}{\unorm{u}\unorm{v}} \end{align*}

Discovery 14.1.

Based on the graph of $y = \cos\theta$ on domain $0 \le \theta \le \pi$ provided below, what can you say about $\udotprod{u}{v}$ in the case that $\theta$ is acute? … obtuse? … right?

$Graph of $y=\cos \theta\text{.}$$

Extending the concept of perpendicular to higher dimensions, vectors $\uvec{u}$ and $\uvec{v}$ are called orthogonal if $\udotprod{u}{v} = 0\text{.}$

Discovery 14.2.

(a)

Can you guess a vector $\uvec{v} = (v_1,v_2)$ that is orthogonal to $\uvec{u} = (1,-3)$ in the plane? Make sure your guess satisfies the definition of orthogonal: you need $\udotprod{u}{v} = 0\text{.}$

(b)

What relationship to your initial guess $\uvec{v}$ will other vectors in the plane that are orthogonal to $\uvec{u}$ have?

Hint

Draw a diagram of your vectors $\uvec{u}$ and $\uvec{v}\text{,}$ both with initial points at the origin. On your diagram, how can you modify your intial guess $\uvec{v}$ geometrically while still maintaining orthogonality with $\uvec{u}\text{?}$

(c)

Turn the pattern of your guess from Task a into a general pattern for vectors in the plane: if $\uvec{u} = (a,b)\text{,}$ then an example of a vector orthogonal to $\uvec{u}$ is

\begin{equation*} \uvec{v} = (\underline{\hspace{1.363636363636364em}},\underline{\hspace{1.363636363636364em}})\text{.} \end{equation*}

Discovery 14.3.

(a)

Draw the vector $\uvec{a} = (3,1)$ in the $xy$-plane with its tail at the origin. Now imagine you were to also draw in every possible scalar multiple of $\uvec{a}$ (positive, negative, zero, fractional, etc.). What geometric shape would these scalar multiples of $\uvec{a}$ trace out? Draw this shape on your diagram.

(b)

Plot the point $Q(4,4)$ on your diagram. On the line defined by $\uvec{a}$ that you drew in the first part of this activity, draw in the point that you think is closest to $Q\text{.}$ Label this point $P\text{.}$ Now draw $\abray{PQ}\text{,}$ and label this vector as $\uvec{n}\text{.}$

What is the relationship between $\uvec{n}$ and the line? What is the value of $\udotprod{n}{a}\text{?}$

(c)

Vector $\abray{OP}$ is parallel to $\uvec{a}\text{,}$ so $\abray{OP}$ is a scalar multiple of $\uvec{a}\text{.}$ Our goal is to determine the scalar $k$ so that the head of $k\uvec{a}$ lies at $P\text{.}$ Complete the triangle in your diagram by drawing in the vector $\uvec{u} = \abray{OQ}\text{.}$

Then express $\uvec{n}$ as a combination of $\uvec{u}$ and $k\uvec{a}\text{.}$

(d)

Substitute your expression for $\uvec{n}$ from Task c into your equation for $\udotprod{n}{a}$ from Task b, and then solve for $k$ as a formula in $\uvec{u}$ and $\uvec{a}\text{.}$

Now complete the general formula:

\begin{equation*} k\uvec{a}=\left(\underline{\hspace{2.727272727272727em}}\right)\uvec{a} \end{equation*}

(where in the brackets you should fill in a formula in the variable letters $\uvec{u}$ and $\uvec{a}\text{,}$ without using their actual numerical components, that describes how to compute $k\uvec{a}$ from $\uvec{u}$ and $\uvec{a}$).

The vector $k\uvec{a}$ in Discovery 14.3 is called the orthogonal projection of $\uvec{u}$ onto $\uvec{a}$, and we write $\uproj{u}{a}$ to mean this vector. It is also sometimes called the vector component of $\uvec{u}$ parallel to $\uvec{a}$. The vector $\uvec{n} = \uvec{u}-\uproj{u}{a}$ is called the vector component of $\uvec{u}$ orthogonal to $\uvec{a}$.

The same problem can be solved in higher dimensions by the same formula for $\uproj{u}{a}\text{.}$

Discovery 14.4.

(a)

Suppose $\uvec{u}$ is orthogonal to $\uvec{a}\text{.}$ What is $\uproj{u}{a}\text{?}$ What is the component of $\uvec{u}$ orthogonal to $\uvec{a}\text{?}$

(b)

Answer the same two questions in the case that $\uvec{u}$ is parallel to $\uvec{a}\text{.}$

Discovery 14.5.

If $\ell$ is the line through the origin and parallel to a vector $\uvec{a}\text{,}$ and $\uvec{u}$ is some other vector, then our construction in Discovery 14.3 guarantees that $\proj_{\uvec{a}} \uvec{u}$ represents the closest point on $\ell$ to the terminal point of $\uvec{u}\text{.}$

The distance between a point and a line is defined as the shortest (i.e. perpendicular) distance between the two. Use the orthogonal projection to come up with a procedure to determine the distance between the line $\ell\colon\,y=x/2$ and the point $Q(2,4)\text{.}$

Discovery 14.6.

The homogeneous linear equation $2 x + 3 y = 0$ defines a line through the origin in $\R^2$ (i.e. the $xy$-plane).

(a)

Recall that a point $(x,y)$ lies on the line if and only if its coordinates satisfy the given equation. Let's consider such a point as the terminal point of the vector $\uvec{x} = (x,y)$ with its initial point at the origin. Does the left-hand side of the equation for the line look like the formula for some quantity related to $\uvec{x}$ and some other vector? Perhaps some quantity that we've been exploring in detail recently?

(b)

In light of the first part of this activity, what does the right-hand side of the equation for the line say about the relationship between a vector $\uvec{x}=(x,y)$ that lies along the line and the other special vector you identified in the previous part?

Discovery 14.7.

The non-homogeneous linear equation $2 x + 3 y = 8$ defines a line through the point $P(1,2)$ in $\R^2\text{.}$

(a)

Draw the line and label the point $P(1,2)\text{.}$ Choose another arbitrary point on the line and label it $Q(x,y)\text{.}$ Draw the vector $\uvec{v} = \abray{PQ}$ along the line. Express the components of $\uvec{v}$ as formulas in $x$ and $y\text{.}$

(b)

Draw the vector $\uvec{n} = (2,3)$ (from the coefficients in the line equation, just as in Discovery 14.6) with its tail at $P\text{.}$ What do you notice about the relationship between this normal vector and the vector $\uvec{v}$ parallel to the line? Express this relationship in terms of the dot product, and then expand out this dot product.

The same sort of analysis can be carried out for a plane in space determined by algebraic equation $a x + b y + c z = d\text{.}$ The coefficients form a normal vector $\uvec{n} = (a,b,c)$ that is perpendicular to the plane (i.e. orthogonal to every vector that is parallel to the plane), and given some specific point $\uvec{x}_0 = (x_0,y_0,z_0)$ that lies on the plane, the plane can be described by the point-normal form $\dotprod{\uvec{n}}{(\uvec{x}-\uvec{x}_0)} = 0\text{.}$

Discovery 14.8.

Consider the planes $\Pi_1\text{,}$ $\Pi_2\text{,}$ and $\Pi_3$ described algebraically below.

\begin{align*} \Pi_1 \amp \colon x - y + 2 z = 2 \amp \Pi_2 \amp \colon 2 x - 2 y + 4 z = 7 \amp \Pi_3 \amp \colon x - y + 3 z = 2 \end{align*}

Use the concept of normal vector to justify the claim that $\Pi_1$ and $\Pi_2$ are parallel, but that $\Pi_3$ is not parallel to either of $\Pi_1$ or $\Pi_2\text{.}$

Orthogonal projection onto a plane in space is a little more complicated, and is likely something you would learn about in a second course in linear algebra. But it's possible to use a different strategy to determine the distance between a point and a plane by using the fact that a plane has one unique normal “direction.”

Discovery 14.9.

(a)

Using the diagram below as inspiration, come up with a procedure to determine the distance $d$ between a point $Q$ and a plane $\Pi\text{.}$

Diagram illustrating procedure to determine the distance between a point and a plane in space.

Hint

Determine a vector that represents an equivalent distance, and then $d$ will be the norm of this vector.

(b)

Come up with a procedure using vectors to determine the distance between parallel planes. Do not assume that either of the planes passes through the origin.

Hint

Find a way to reduce this problem to the problem in the first part of this activity.