ilovepdf_merged (6).pdf). Content covers sections 33.1–33.x.Extend the plane to three dimensions with mutually perpendicular axes OX, OY, OZ. A point is an ordered triple (x,y,z).
Mid-point and section formulae extend by applying the same ratios to each coordinate. Origin O(0,0,0); axes intercepts analogous to 2D.
If a line makes angles α,β,γ with the axes, direction cosines l=cos α, m=cos β, n=cos γ satisfy l² + m² + n² = 1. Direction ratios a,b,c are proportional to l,m,n.
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Introduction to Three Dimensional Geometry.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: 13. Find the vector equation of a line passing through the points (3, –2, –5) and
Memorise and apply: Distance of a point
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
4 question(s) · Sources: Board-style DR, Board-style midpoint 3D, Oct 2024 / Sample QP 2024, Sample QP 2024
PYQ1. The distance between the points (3, 5, −1) and (9, 2, −4) is:
Model Answer
1) 3√6 units
Explanation
1) 3D distance √[(9−3)²+(2−5)²+(−4−(−1))²]=√(36+9+9)=√54=3√6.
PYQ2. P divides the join of A(2,−5,2) and B(−3,5,2) internally in ratio 1:4. Find coordinates of P.
Model Answer
1) P = (1, −3, 2)
Explanation
1) Section formula: x=(1·(−3)+4·2)/(1+4)=(−3+8)/5=1
2) y=(1·5+4·(−5))/5=(5−20)/5=−3
3) z=(1·2+4·2)/5=10/5=2.
PYQ3. Direction ratios 2, −1, 2. Find the direction cosines.
Model Answer
1) (2/3, −1/3, 2/3)
Explanation
1) Magnitude √(4+1+4)=3; DC = (2/3, −1/3, 2/3).
PYQ4. Find the mid-point of (2,0,−1) and (4,2,5).
Model Answer
1) (3, 1, 2)
Explanation
1) Average coordinates: ((2+4)/2,(0+2)/2,(−1+5)/2)=(3,1,2).
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
Distance between A(1,2,3) and B(1,2,7).
1) √(0+0+16)=4.
Answer: 4
Only z differs by 4.
Same x,y → vertical segment length 4.
3D distance reduces to |Δz| here.
Do not write √(1+2+3).
Mid-point of (2,0,−1) and (4,2,5).
1) ((2+4)/2,(0+2)/2,(−1+5)/2)=(3,1,2).
Answer: (3, 1, 2)
Section formula with equal weights.
Average each coordinate.
Same as 2D but three components.
Sign of z: (−1+5)/2=2.
Direction ratios 2,−1,2. Find direction cosines.
1) √(4+1+4)=3.
2) l=2/3, m=−1/3, n=2/3.
Answer: (2/3, −1/3, 2/3)
Normalise the direction ratio vector.
Divide by the magnitude 3.
l²+m²+n²=1 check: 4/9+1/9+4/9=1.
Keep the minus on m.
If two direction cosines are 1/√2 and 0, find the third (two possibilities).
1) l²+m²+n²=1 ⇒ 1/2 + 0 + n²=1 ⇒ n²=1/2 ⇒ n=±1/√2.
Answer: ±1/√2
Identity for direction cosines.
Squares sum to 1.
Two orientations of the line’s sense.
Do not forget ±.
Point dividing join of (1,0,0) and (0,1,0) in 1:1.
1) Mid-point ((1+0)/2,(0+1)/2,(0+0)/2)=(1/2,1/2,0).
Answer: (1/2, 1/2, 0)
Internal section 1:1 is mid-point in 3D.
Average the endpoints.
Lies in the plane z=0.
Ratio 1:1 special case.
Are A(1,2,3), B(2,3,4), C(3,4,5) collinear?
1) AB⃗=(1,1,1), AC⃗=(2,2,2)=2(1,1,1).
2) Direction ratios proportional ⇒ collinear.
Answer: Yes, collinear
Vectors AB and AC are scalar multiples.
They lie on a straight line in equal steps.
Distance AB+BC=AC also works: each step √3, total 2√3.
Proportional components required.
Find the distance between (0, 0, 0) and (2, 3, 6).
1) √(4+9+36)=√49=7.
Answer: 7
Three-dimensional distance formula.
2-3-6 gives length 7.
From origin: √(x²+y²+z²).
Check each algebraic step carefully.
Find the mid-point of (2, 0, −4) and (6, 4, 2).
1) ((2+6)/2, (0+4)/2, (−4+2)/2) = (4, 2, −1).
Answer: (4, 2, −1)
Average each coordinate.
Same as 2D mid-point with a z-component.
Internal section with equal weights.
Check each algebraic step carefully.
Direction ratios are 1, 2, 2. Find the direction cosines.
1) r = √(1+4+4)=3.
2) DC = (1/3, 2/3, 2/3).
Answer: (1/3, 2/3, 2/3)
Divide DR by their magnitude.
Unit vector in that direction.
Check 1/9+4/9+4/9=1.
Check each algebraic step carefully.
Find the distance of P(1, −2, 2) from the origin.
1) OP=√(1+4+4)=√9=3.
Answer: 3
Distance from origin in 3D.
Same as magnitude of position vector.
Signs square away.
Check each algebraic step carefully.