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Discovery guide 15.1 Discovery guide

Discovery 15.1.

Begin with a set of \(xy\)-axes. Draw the vector \(\uvec{x}_0 = (3,0)\) with its tail at the origin, and then draw the vector \(\uvec{p} = (2,1)\) with its tail at the head of \(\uvec{x}_0\text{.}\)

(a)

Consider the expression \(\uvec{x} = \uvec{x}_0 + t \uvec{p}\) in the parameter t. Think of \(\uvec{x}\) as a variable vector: using different values of \(t\text{,}\) \(\uvec{x}\) evaluates to different vectors. Draw the vector \(\uvec{x}\) for \(t=1\) on your diagram with its tail at the origin and using a dashed line for the shaft of the arrow. Then do the same for \(t=2\text{,}\) \(t=-1\text{,}\) \(t=1/2\text{,}\) \(t=-3\text{.}\)

(b)

Suppose you continued sketching in the different possible \(\uvec{x}\) vectors forever, using every possible value for the parameter \(t\text{.}\) What shape would be traced out by all of the points at the heads of the different versions of \(\uvec{x}\text{?}\)

Discovery 15.2.

The equation \(x - 2 y = 3\) defines a line \(\ell\) in \(\R^2\text{.}\) We can also view this equation as a system of linear equations. Its solution requires one parameter.

(a)

Set \(y = t\) and then compute the parametric equation for \(x\text{.}\) Set \(\uvec{x}\) to be the variable vector \(\uvec{x} = \left[\begin{smallmatrix}x\\y\end{smallmatrix}\right]\text{.}\) Fill in the vectors at below. Then compare with Discovery 15.1.

\begin{equation*} \uvec{x} \;\;=\;\;\begin{bmatrix}x\\y\end{bmatrix} \;\;=\;\;\begin{bmatrix}\hphantom{3+2t}\\t\end{bmatrix} \;\;=\;\;\begin{bmatrix}\hphantom{3}\\\hphantom{0}\end{bmatrix} \;\;+\;\; t\begin{bmatrix}\hphantom{2}\\\hphantom{1}\end{bmatrix} \end{equation*}
(b)

Use the line equation \(x - 2 y = 3\) to verify that the point \((4,1/2)\) lies on \(\ell\text{.}\) Then determine the value of the parameter \(t\) so that \(\uvec{x} = (4,1/2)\text{.}\)

Discovery 15.3.

Consider the two planes

\begin{align*} \Pi_1\amp\colon\;\; 2 x - y + 5 z = -5, \amp \Pi_2\amp\colon\;\; x + 2 y - 5 z = 10 \end{align*}

in \(\R^3\text{.}\)

(a)

Verify that \(\Pi_1\) and \(\Pi_2\) are not parallel.

Hint

Compare their normal vectors.

(b)

Two nonparallel planes must intersect in a line. Describe the line of intersection of \(\Pi_1\) and \(\Pi_2\) in the form \(\uvec{x} = \uvec{x}_0 + t \uvec{p}\text{.}\)

Hint

Any point in the intersection must lie on both planes at once. That is, any point in the intersection must be a solution to the system of equations formed by the two plane equations.

Discovery 15.4.

The equation \(x - y + 5 z = -5\) defines a plane in \(\R^3\text{.}\) We can also view this equation as a system of linear equations.

(a)

Similarly to Discovery 15.2, determine vectors \(\uvec{x}_0\text{,}\) \(\uvec{p}_1\text{,}\) and \(\uvec{p}_2\) so that

\begin{equation*} \uvec{x} = \left[\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right] = \uvec{x}_0 + s\uvec{p}_1 + t\uvec{p}_2 \end{equation*}

describes all solutions to the equation (and hence all points on the plane).

(b)

Use the plane's equation \(x - y + 5 z = -5\) to verify that the point \((1,1,-1)\) lies on the plane.

Then determine the values of the parameters \(s\) and \(t\) so that the formula

\begin{equation*} \uvec{x} = \uvec{x}_0 + s\uvec{p}_1 + t\uvec{p}_2 \end{equation*}

results in this point \(\uvec{x} = (1,1,-1)\text{.}\)

Discovery 15.5.

Draw a grid over the \(xy\)-plane, with a vertical line at each integer value of \(x\) and a horizontal line at each integer value of \(y\text{.}\) Then draw \(\uvec{e}_1\) and \(\uvec{e}_2\) on your diagram.

What does the decomposition \((3,2) = 3\uvec{e}_1 + 2\uvec{e_2}\) look like on your grid?

How about \((-1,2) = (-1)\uvec{e}_1 + 2\uvec{e}_2\text{?}\)

How about \((3/2,-2) = (3/2)\uvec{e}_1 + (-2)\uvec{e}_2\text{?}\)

Discovery 15.6.

Draw a “grid” over the \(xy\)-plane as follows: at each integer value along the \(x\)-axis, draw both a vertical line and a slant line parallel to the line \(y=x\text{.}\) Then draw \(\uvec{u} = (1,1)\) and \(\uvec{e}_2\) on your diagram.

What does the decomposition \((3,2) = 3\uvec{u} + (-1)\uvec{e}_2\) look like on your grid?

How about \((-1,2) = (-1)\uvec{u} + 3\uvec{e}_2\text{?}\)

How about \((3/2,-2) = (3/2)\uvec{u} + (-7/2)\uvec{e}_2\text{?}\)

Discovery 15.7.

The set of all solutions to the homogeneous equation \(x - 2 y + 3 z = 0\) forms a plane in \(\R^3\text{.}\) We can solve this equation by assigning parameters \(y=s\) and \(z=t\text{,}\) so that all solutions can be described parametrically by

\begin{equation*} (x,y,z) = s(2,1,0) + t(-3,0,1) \text{.} \end{equation*}

Discuss how the vectors \(\uvec{p}_1 = (2,1,0)\) and \(\uvec{p}_2 = (-3,0,1)\) create a “grid” on the plane defined by \(x - 2 y + 3 z = 0\text{,}\) similarly to the grids you worked with in Discovery 15.5 and Discovery 15.6.

Discovery 15.8.

Determine the point of intersection of the line \(\ell\text{,}\) described parametrically below left, and the plane \(\Pi\text{,}\) described algebraically below right.

\begin{align*} \ell\amp\colon\;\; \uvec{x} = (2,0,3) + t(-1,1,1) \amp \Pi\amp\colon\;\; 2 x + y - 3 z = 7 \end{align*}
Hint

The point of intersection is simultaneously on the line and on the plane.

Discovery 15.9.

Set up a system of equations whose solution is the point of intersection of the line \(\ell\) and the plane \(\Pi\text{,}\) described parametrically below.

\begin{align*} \ell\amp\colon\;\; \uvec{x} = (2,0,3) + t(-1,1,1) \amp \Pi\amp\colon\;\; \uvec{x} = (3,1,0) + r(1,1,1) + s(3,0,2) \end{align*}
Hint

The point of intersection is simultaneously on the line and on the plane.