Discover Linear Algebra: A first course in linear algebra

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

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

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

Plot the point $Q(4,4)$ on your diagram. On the line defined by $\mathbf{a}$ 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 $\overrightarrow{PQ}\text{,}$ and label this vector as $\mathbf{n}\text{.}$

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

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

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

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

Suppose $\mathbf{u}$ is orthogonal to $\mathbf{a}\text{.}$ What is ${\mathrm{proj}}_{\mathbf{a}}\mathbf{u}\text{?}$ What is the component of $\mathbf{u}$ orthogonal to $\mathbf{a}\text{?}$

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

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 $\mathbf{x}=(x,y)$ 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 $\mathbf{x}$ and some other vector? Perhaps some quantity that we’ve been exploring in detail recently?

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 $\mathbf{x}=(x,y)$ that lies along the line and the other special vector you identified in the previous part?

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 $\mathbf{v}=\overrightarrow{PQ}$ along the line. Express the components of $\mathbf{v}$ as formulas in $x$ and $y\text{.}$

Draw the vector $\mathbf{n}=(2,3)$ (from the coefficients in the line equation, just as in Discovery 13.6) with its tail at $P\text{.}$ What do you notice about the relationship between this normal vector and the vector $\mathbf{v}$ parallel to the line? Express this relationship in terms of the dot product, and then expand out this dot product.

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

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.