Lab
Vectors and Matrices
Choose what to explore. With vectors, drag the heads of u and v or type their components, and pick whether to draw u + v, k u or u − v. Before you look, say where the new vector's head will land, then check. With a matrix, type the four entries of A or start from a rotation, a reflection or a dilation. The dashed square is the unit square and the shaded parallelogram is its image under A, so the sides of the image are the columns of A. Drag v and watch A v follow. With B after A, the image under A is carried on by B, and the lab shows that the single matrix BA does both at once. Every value is exact: fractions stay fractions and roots stay roots. These topics are in Unit 4, which is not assessed on the AP Exam.
Challenges
- With u = ⟨3, 1⟩ and v = ⟨1, 2⟩, find u + v and |u + v| by hand, then draw u + v tip to tail and check. Is |u + v| equal to |u| + |v|? Why not?
- Keep u = ⟨3, 1⟩ and find a vector v that is perpendicular to it. What is u ⋅ v, and what angle does the lab give between u and v?
- Set k to −1/2 and draw k u. Which way does it point compared with u, and how long is it compared with |u|?
- Choose the matrix mode and the rotation by θ, and set θ to π/2. What are the columns of A? Where does v = ⟨3, −1⟩ go? Check the formula the lab gives for A⟨x, y⟩.
- Type the matrix with first row 2, 1 and second row 4, 2. Before you look, find det(A). What happens to the unit square, and why does the lab say A has no inverse?
- Choose the reflection over the line y = x. Say what det(A) is before you look, then check. Find A⁻¹ and explain why it is A itself.
- Choose a dilation by k = 2. What is the area of the image of the unit square, and how does det(A) give it?
- Choose B after A, with A the rotation by π/2 and B the reflection over the x-axis. Where does BA send ⟨1, 0⟩ and ⟨0, 1⟩? Now make A the reflection over the x-axis and B the rotation by π/2. Where does this BA send ⟨1, 0⟩ and ⟨0, 1⟩? Compare both with the reflection over the line y = x. Are the two composites the same?