ilovepdf_merged (6).pdf). Content covers sections 36.1–36.x.A line in space is fixed by a point and a direction, or by intersection of two planes. Use vector and Cartesian symmetric forms.
Line through point with position vector a⃗ and parallel to b⃗: r⃗ = a⃗ + λ b⃗.
Angle between two lines: cos θ = |b₁·b₂| / (|b₁||b₂|). Skew lines neither intersect nor are parallel; distance formulas use the triple product as in the textbook.
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Straight Line.
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: = OP
Memorise and apply: 36.1.3 EQUATION OF THE LINE PASSING THROUGH TWO
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 of join, Board-style angle, Board-style parametric, Board-style symmetric form
PYQ1. Write the symmetric equations of the line through (1,2,3) parallel to î+2ĵ−2k̂.
Model Answer
1) (x−1)/1 = (y−2)/2 = (z−3)/(−2)
Explanation
1) Point–direction form with DR (1,2,−2).
PYQ2. Find the angle between lines with directions (1,0,0) and (1,1,0).
Model Answer
1) π/4
Explanation
1) cos θ = |1|/(1·√2)=1/√2 ⇒ θ=π/4.
PYQ3. If (x−1)/2=(y+1)/(−1)=z/3=λ, find the point for λ=1.
Model Answer
1) (3, −2, 3)
Explanation
1) x=1+2λ, y=−1−λ, z=3λ; λ=1 ⇒ (3,−2,3).
PYQ4. Direction ratios of the line joining (1,0,0) and (1,0,2).
Model Answer
1) (0, 0, 1) [or (0,0,2)]
Explanation
1) Difference of coordinates (0,0,2).
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.
Write the symmetric equations of the line through (1,2,3) parallel to i+2j−2k.
1) (x−1)/1 = (y−2)/2 = (z−3)/(−2).
Answer: (x−1)/1 = (y−2)/2 = (z−3)/(−2)
Point and direction ratios give symmetric form.
Subtract point, divide by DR.
Can multiply DR by −1 consistently.
Keep the point coordinates with minus.
Vector equation of line through A with OA⃗=i+j and parallel to b⃗=k.
1) r⃗ = (i+j) + λ k.
Answer: r⃗ = i + j + λ k
Standard vector equation of a line.
Start at A, add multiples of direction.
λ ∈ ℝ.
Do not add λ to each basis wrongly without direction.
Angle between lines with directions (1,0,0) and (1,1,0).
1) cosθ=|1|/(1·√2)=1/√2 ⇒ θ=π/4.
Answer: π/4
Formula for angle between direction vectors.
Dot product 1, magnitudes 1 and √2.
Acute angle chosen via absolute value.
Not 45° without stating unit if radians expected — π/4 preferred.
Are lines r=i+λ(2j) and r=j+μ(−4i+0j) parallel? Wait directions (0,2,0) and (−4,0,0).
1) Directions (0,2,0) and (−4,0,0) are not scalar multiples (one has only j, other only i).
2) Not parallel.
Answer: Not parallel
Parallel requires proportional direction vectors.
i-direction vs j-direction → perpendicular actually.
Dot product 0 ⇒ perpendicular lines (if they intersect or as directions).
Proportional means each component scales the same.
From (x−1)/2=(y+1)/−1=(z)/3=λ, find the point when λ=1.
1) x=1+2=3; y=−1−1=−2; z=0+3=3.
2) Point (3,−2,3).
Answer: (3, −2, 3)
Parametric equations from symmetric form.
Plug λ=1 into each coordinate.
y₀=−1 because y−(−1).
Sign in y: −1 + (−1)λ.
Direction ratios of line joining (1,0,0) and (1,0,2).
1) (0,0,2) or (0,0,1).
Answer: (0, 0, 1) [or (0,0,2)]
Subtract coordinates of the two points.
Only z changes.
Any non-zero multiple is valid DR.
Not (1,0,2).
Write the vector equation of the line through (1,0,0) parallel to j.
1) r = i + λ j (or r = <1,0,0> + λ<0,1,0>).
Answer: r = <1, 0, 0> + λ <0, 1, 0>
Point + scalar times direction.
Parallel to y-axis through (1,0,0).
λ real parameter.
Check each algebraic step carefully.
Write symmetric equations of the line through (2,3,4) with DR <1,0,0>.
1) (x−2)/1 = (y−3)/0 = (z−4)/0 is written carefully: y=3, z=4, x free.
2) Or x=2+λ, y=3, z=4.
Answer: x = 2 + λ, y = 3, z = 4
Zero direction components mean fixed coordinates.
Line parallel to x-axis.
Avoid dividing by zero in symmetric form.
Check each algebraic step carefully.
Are directions <1,2,3> and <2,4,6> parallel?
1) Second = 2 × first ⇒ yes, parallel directions.
Answer: Yes
Parallel direction vectors are scalar multiples.
Same or opposite sense.
Lines may still be skew if not coplanar.
Check each algebraic step carefully.
Does (3,2,1) lie on r = <1,2,1> + λ<2,0,0>?
1) y=2, z=1 match for all λ.
2) x=1+2λ=3 ⇒ λ=1.
3) Yes — when λ=1.
Answer: Yes (λ = 1)
Solve for λ consistently in all coordinates.
y and z already match the line.
If y failed, point off the line.
Check each algebraic step carefully.