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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-11-LIN-01
Equation of a Line in Space — Assignment
Chapter: Three-Dimensional Geometry
Topic: Equation of a Line in Space
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
The vector equation of a line is:
  • A.$\vec r=\vec a+\lambda\vec b$
  • B.$\vec r\cdot\vec n=d$
  • C.$\vec r=\vec a\times\vec b$
  • D.$|\vec r|=1$
2.
In $\dfrac{x-1}{2}=\dfrac{y+3}{-1}=\dfrac{z}{4}$, the direction ratios are:
  • A.$(1,-3,0)$
  • B.$(2,-1,4)$
  • C.$(1,3,0)$
  • D.$(2,1,4)$
3.
A point on $\dfrac{x-1}{2}=\dfrac{y+3}{-1}=\dfrac{z}{4}$ is:
  • A.$(2,-1,4)$
  • B.$(1,-3,0)$
  • C.$(1,3,0)$
  • D.$(0,0,0)$
4.
The line through $(1,2,3),(2,4,5)$ has DRs:
  • A.$(1,2,2)$
  • B.$(3,6,8)$
  • C.$(1,1,1)$
  • D.$(2,4,5)$
5.
In Cartesian form, the denominators give the:
  • A.points
  • B.direction ratios
  • C.angles
  • D.intercepts
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Write the vector equation of the line through $(1,0,2)$ with direction $\hat i+\hat j-\hat k$.
7.
Write the Cartesian equation of the line through $(2,-1,3)$ with DRs $1,2,2$.
8.
Write the Cartesian equation of the line through $(1,2,3)$ and $(2,4,5)$.
9.
From $\dfrac{x-1}{2}=\dfrac{y+3}{-1}=\dfrac{z}{4}$, write a point and the DRs.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Find the vector equation of the line through $(2,-1,1)$ and $(4,1,3)$.
11.
Find the Cartesian equation of the line through the origin with DRs $3,-1,2$.
12.
Does the point $(3,2,1)$ lie on $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z}{1}$?
13.
Convert $\vec r=(\hat i+\hat j)+\lambda(2\hat i-\hat j+\hat k)$ to Cartesian form.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Find the vector and Cartesian equations of the line through $(1,2,3)$ parallel to $\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{6}$.
15.
Show that the line through $(1,-1,2),(3,4,-2)$ is perpendicular to the line through $(0,3,2),(3,5,6)$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $\vec r=\vec a+\lambda\vec b$
  2. (B) $(2,-1,4)$
  3. (B) $(1,-3,0)$
  4. (A) $(1,2,2)$
  5. (B) direction ratios
Section B — Short Answer (2 marks)
  1. $\vec r=(\hat i+2\hat k)+\lambda(\hat i+\hat j-\hat k)$.
  2. $\dfrac{x-2}{1}=\dfrac{y+1}{2}=\dfrac{z-3}{2}$.
  3. $\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{2}$.
  4. Point $(1,-3,0)$; DRs $(2,-1,4)$.
Section C — Short Answer (3 marks)
  1. $\vec r=(2\hat i-\hat j+\hat k)+\lambda(\hat i+\hat j+\hat k)$.
  2. $\dfrac{x}{3}=\dfrac{y}{-1}=\dfrac{z}{2}$.
  3. Yes.
  4. $\dfrac{x-1}{2}=\dfrac{y-1}{-1}=\dfrac{z}{1}$.
Section D — Long Answer (5 marks)
  1. $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+6\hat k)$; $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{6}$.
  2. Perpendicular (dot product $=0$).
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