⚡ Key Formulas at a Glance
Distance Formula 3D
d = √[(x2−x1)²+(y2−y1)²+(z2−z1)²]
Section Formula 3D
P = ((mx2+nx1)/(m+n), ...)
Distance from XY-plane
|z|
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1Exercise 12.1
Find the distance between the points (2, 3, 5) and (4, 3, 1).
▶ Show Detailed Solution
d = √[(4-2)² + (3-3)² + (1-5)²]
= √[4 + 0 + 16]
= √20 = 2√5 units
2Exercise 12.1
Verify that the points (0, 7, -10), (1, 6, -6) and (4, 9, -6) form an isosceles triangle.
▶ Show Detailed Solution
AB = √[(1-0)² + (6-7)² + (-6+10)²] = √[1 + 1 + 16] = √18
BC = √[(4-1)² + (9-6)² + (-6+6)²] = √[9 + 9 + 0] = √18
CA = √[(4-0)² + (9-7)² + (-6+10)²] = √[16 + 4 + 16] = √36 = 6
AB = BC = √18, so the triangle is isosceles.
3Exercise 12.1
Find the coordinates of the point which divides the line joining A(1, -2, 3) and B(3, 4, -5) in the ratio 2 : 3 internally.
▶ Show Detailed Solution
Section formula (internal), m : n = 2 : 3:
P = ((m x₂ + n x₁)/(m+n), (m y₂ + n y₁)/(m+n), (m z₂ + n z₁)/(m+n))
= ((2×3 + 3×1)/5, (2×4 + 3×(-2))/5, (2×(-5) + 3×3)/5)
= ((6+3)/5, (8-6)/5, (-10+9)/5)
= (9/5, 2/5, -1/5)
4Exercise 12.1
What are the coordinates of the centroid of the triangle with vertices (1, 2, 3), (4, 5, 6) and (7, 8, 9)?
▶ Show Detailed Solution
Centroid G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
= ((1+4+7)/3, (2+5+8)/3, (3+6+9)/3)
= (4, 5, 6)
5Exercise 12.1
Find the direction cosines of a line that makes equal angles with the coordinate axes.
▶ Show Detailed Solution
Let the equal angle be α with each axis.
cos²α + cos²α + cos²α = 1
3cos²α = 1 ⇒ cos α = ±1/√3
Direction cosines = (±1/√3, ±1/√3, ±1/√3)
For a line, all have the same sign.
So direction cosines = (1/√3, 1/√3, 1/√3).