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Precalculus 9-12 / Week 11 / Wednesday
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Week 11 · Matrices

Wednesday

Boathouse Lab: the boat outline
// The lane card grid
⏱ about 20 min

Wednesday: Boathouse Lab, the Boat Outline

Comet traces the cardboard hull on graph paper. "Corners at (0, 0), (4, 0), (3, 1) and (0, 1)."

"Area 3.5 squares," Wren says, counting. "Now a matrix moves every corner."

He writes [2 0; 0 1]. "Multiply each corner by this. (4, 0) becomes (8, 0)."

Comet plots the four new corners and cuts a second hull from cardboard with scissors. "Twice as long. Area 7."

Nova projects both hulls, blue and red. "Would you like a hint? Compare the area with ad - bc."

"2 × 1 - 0 × 0 = 2," Wren says. "The area doubled and the determinant is 2."

"Try the flip," Comet says, writing [1 0; 0 -1]. "What do you notice about its determinant?"

"-1. Negative, because the hull turned over. The area factor is the absolute value, 1."

What you need

  • Graph paper, a pencil, a ruler, a sheet of cardboard, scissors and your Boathouse Log.
  • Four colors of pencil or pen for the four transformed hulls.
Safety first
Cut the cardboard with scissors only, sitting down, with the points turned away from you. Nothing sharp is used.
A grown-up is nearby. Keep cardboard scraps off the floor.
The lab stays at the table. On the water the crew wears life vests, and a grown-up is on the dock.
Nothing heavy is lifted alone.

A matrix moves a point

Write a point (x, y) as a column, [x; y]. A 2 by 2 matrix times that column is a new column, a new point.

[2 0; 0 1] × [x; y] = [2x + 0y; 0x + 1y] = [2x; y]. Every x doubles and every y stays. That is a stretch across.

[1 0; 0 -1] × [x; y] = [x; -y]. Every point flips over the x-axis.

[1 1; 0 1] × [x; y] = [x + y; y]. Points slide across by their height. That is a shear.

Apply the matrix to each corner of a shape and join the new corners. The whole shape has been transformed.

The blue hull outline and its red image under [2 0; 0 1], twice as long and the same height.

Run the lab

  1. Trace the hull on graph paper with corners (0, 0), (4, 0), (3, 1), (0, 1). Count its area in squares.
  2. Pick a matrix from the table. Multiply each corner by it to get four new corners.
  3. Plot the new corners in a new color and join them. Count the new area, half squares included.
  4. Work out the determinant ad - bc. Compare its absolute value with new area ÷ old area.
  5. Repeat for all four matrices. Cut the stretched hull from cardboard with scissors and lay it on the drawing.

The crew's lab log

The hull's area is 3.5 squares. Here are the crew's four transformations, with every area counted on the grid.

TransformationMatrixDeterminantArea factorNew area
Stretch across[2 0; 0 1]227
Flip over the x-axis[1 0; 0 -1]-113.5
Shear[1 1; 0 1]113.5
Stretch and turn[2 1; 1 -1]-3310.5
The blue hull and its red image under [2 1; 1 -1], stretched, turned and flipped over.

Every row agrees: new area = old area × |determinant|. Under [2 1; 1 -1] the hull turned over and tripled, so the determinant is -3.

The shear [1 1; 0 1] leans the hull but keeps its area. Its determinant is 1, and 3.5 × 1 = 3.5.

A negative determinant means the shape turned over, like the flipped hull. The size of the number is still the area factor.

READ THE LAB LOG
  • Read the question.
  • Tap your answer.
A blue shape with vertices (0, 0), (3, 1) and its red image under [2 0; 0 1], vertices (0, 0), (6, 1)Apply the stretch [2 0; 0 1] to the hull corner ⟨3, 1⟩. Where does it land?
A blue shape with vertices (0, 0), (3, 1) and its red image under [1 1; 0 1], vertices (0, 0), (4, 1)Apply the shear [1 1; 0 1] to the corner ⟨3, 1⟩. Where does it land?
A blue shape with vertices (0, 0), (1, 0), (1, 1), (0, 1) and its red image under [2 1; 1 -1], vertices (0, 0), (2, 1), (3, 0), (1, -1)The matrix [2 1; 1 -1] transformed the hull. By what factor did it multiply the area?
The hull's area is 3.5 squares. After the stretch [2 0; 0 1], what is the new area, in squares?
What is the determinant of the flip [1 0; 0 -1]? Type the number.
WHY THIS EXERCISEA determinant of -1 says the area is unchanged and the shape turned over. Both facts live in one number.
What the lab showsTrue or false?
Multiplying each corner by the matrix transforms the whole shape.?
3.5 × 2 = 7?
A shear changes the area of the hull.?
A negative determinant means the shape was turned over.?
The area factor is the determinant itself, sign included.?
WHY THIS EXERCISEThe lab turns a formula into cardboard you can hold: the determinant is how much the hull grew.
Draw the hull and its four images in four colors. Write each matrix and its determinant beside its image.

Careful lab work. Tomorrow the determinant decides whether a matrix can be undone, and a system of equations becomes one matrix sentence.

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