Space coordinates, distance, midpoint and the first vectors
Answers are for parents and teachers.
1
Write the distances from (−5, 12, −3) to (a) the xy-plane, (b) the x-axis, (c) the z-axis.
(a) |−3| = 3. (b) √(144 + 9) = √153 = 3√17. (c) √(25 + 144) = 13.
2
Find the distance between A(−1, 4, 2) and B(5, −2, 5).
Differences 6, −6, 3: √(36 + 36 + 9) = 9.
3
Show that P(3, 1, 0), Q(1, 0, 2) and R(2, 0, 3) form a right-angled triangle, and find its area.
PQ² = 4 + 1 + 4 = 9, QR² = 1 + 0 + 1 = 2, PR² = 1 + 1 + 9 = 11. Since 9 + 2 = 11 the right angle is at Q. Area ½ × 3 × √2 = 3√2/2.
4
The end A is (−2, 5, 0) and the midpoint of AB is M(4, −1, 2). Find the other end B.
B = 2M − A = (8 + 2, −2 − 5, 4 − 0) = (10, −7, 4).
5
For which values of k does the vector (2, k, −3) have norm 7?
4 + k² + 9 = 49, so k² = 36 and k = 6 or k = −6.