Cartesian coordinates in space
Answers are for parents and teachers.
1
Find the distance between A(−1, 2, 3) and B(3, −2, 5) and the midpoint of AB.
AB = √(16 + 16 + 4) = √36 = 6. Midpoint: (1, 0, 4).
2
For the triangle A(0, 0, 0), B(3, 4, 0), C(0, 0, 12) find the sides and show that it is right-angled.
AB = 5, AC = 12, BC = √(9 + 16 + 144) = 13; 5² + 12² = 13², so the right angle is at A.
3
Write the equation of the sphere whose diameter has endpoints (5, −2, 0) and (−1, 4, 2).
The centre is the midpoint (2, 1, 1); R² = 3² + 3² + 1² = 19: (x − 2)² + (y − 1)² + (z − 1)² = 19.
4
Decide whether (1, 2, 3) lies inside, on the surface of, or outside the ball (x − 2)² + y² + (z − 1)² ≤ 9.
1 + 4 + 4 = 9: it lies on the sphere (the boundary of the ball).
5
Find the point P that divides the segment A(0, 0, 0), B(8, 4, 12) in the ratio AP : PB = 3 : 1.
λ = 3: P = (3 · 8/4, 3 · 4/4, 3 · 12/4) = (6, 3, 9).