Lesson 5 of 11 · 7 min
Two points and the section formula
NCERT §10.5.2–10.5.3
The drone flies a straight survey leg from A(2, 0, 4) to B(8, 6, 10), and the camera must fire two-thirds of the way along. At what point should the drone be?
The lesson in notes
In short
The vector from P₁(x₁, y₁, z₁) to P₂(x₂, y₂, z₂) is P₁P₂ = OP₂ − OP₁ = (x₂ − x₁)î + (y₂ − y₁)ĵ + (z₂ − z₁)k̂: head minus tail. Its magnitude is the distance formula.
Worked: from P(2, 3, 0) to Q(−1, −2, −4), PQ = −3î − 5ĵ − 4k̂.
If R divides PQ internally in the ratio m : n (PR : RQ = m : n) and P, Q have position vectors a, b, then OR = (mb + na)/(m + n).
If R divides PQ externally in the ratio m : n (R outside the segment, m ≠ n), then OR = (mb − na)/(m − n).
The mid-point (m = n) has position vector (a + b)/2.
Worked: for P = î + 2ĵ − k̂ and Q = −î + ĵ + k̂, the 2 : 1 internal point is (−î + 4ĵ + k̂)/3 and the 2 : 1 external point is −3î + 3k̂.
Worked: A(2, −1, 1), B(1, −3, −5), C(3, −4, −4) give |AB|² = 41, |BC|² = 6, |CA|² = 35, and 41 = 6 + 35, so ABC is right-angled (at C).