Discover Linear Algebra

(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?

(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

(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:

(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 $\uvec{a}$).

(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 $\uvec{a}\text{.}$

(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?

(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.

(a)

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

(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.