Transformations and chapter review
Answers are for parents and teachers.
1
Under a translation the point (2, −1, 3) goes to (5, 0, 1). Which vector is it and where does the origin go?
p = (3, 1, −2). The origin goes to (3, 1, −2).
2
Write the points symmetric to A(3, −2, 6) in the plane Oxy, the axis Ox and the origin.
Oxy: (3, −2, −6); Ox: (3, 2, −6); origin: (−3, 2, −6).
3
Find the perimeter of triangle A(1, 2, 3), B(3, 2, 1), C(3, 4, 3) and its type.
AB = √8, BC = √(0 + 4 + 4) = √8, AC = √(4 + 4 + 0) = √8 = 2√2; perimeter 6√2; equilateral.
4
Find the cosine of the angle between a(1, 2, −2) and b(2, 1, 2).
a · b = 2 + 2 − 4 = 0, so cos φ = 0 and φ = 90°.
5
Under the homothety with centre (0, 0, 0) and k = 3 find the images of A(1, −2, 4) and B(2, 0, 1) and the new length of AB.
A′(3, −6, 12), B′(6, 0, 3). AB = √(1 + 4 + 9) = √14, A′B′ = 3√14.