Vidaara.orgClass 12 · Mathematics
CodeVID-M12-11-ANG-01
Angle Between Lines & Shortest Distance — Assignment
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.
Two lines are perpendicular when their direction vectors satisfy:
- A.$\vec b_1\times\vec b_2=\vec0$
- B.$\vec b_1\cdot\vec b_2=0$
- C.$\vec b_1=\vec b_2$
- D.$|\vec b_1|=|\vec b_2|$
2.
The cosine of the angle between two lines uses:
- A.cross product
- B.$\dfrac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|}$
- C.the sum
- D.the determinant
3.
Skew lines intersect when the shortest distance is:
- A.$1$
- B.$0$
- C.$\infty$
- D.negative
4.
Lines with directions $2\hat i+4\hat j+6\hat k$ and $\hat i+2\hat j+3\hat k$ are:
- A.perpendicular
- B.parallel
- C.skew
- D.intersecting
5.
The shortest-distance formula for skew lines uses:
- A.$\vec b_1+\vec b_2$
- B.$\vec b_1\times\vec b_2$
- C.$\vec b_1\cdot\vec b_2$
- D.$|\vec b_1|$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the angle between lines with directions $\hat i+\hat j+\hat k$ and $\hat i-\hat j+\hat k$.
7.
Show that the directions $\hat i+\hat j$ and $\hat i-\hat j$ are perpendicular.
8.
Are the directions $2\hat i+4\hat j+6\hat k$ and $\hat i+2\hat j+3\hat k$ parallel?
9.
Find the cosine of the angle between $\hat i$ and $\hat i+\hat j$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the angle between lines with directions $(1,2,2)$ and $(2,2,1)$.
11.
Find the angle between directions $(1,1,2)$ and $(2,1,-1)$.
12.
Show that lines with directions $(1,2,3)$ and $(-1,-2,-3)$ are parallel.
13.
Find $\lambda$ so that lines with directions $(2,\lambda,3)$ and $(1,-1,2)$ are perpendicular.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Find the shortest distance between $\vec r=(\hat i+2\hat j+\hat k)+\lambda(\hat i-\hat j+\hat k)$ and $\vec r=(2\hat i-\hat j-\hat k)+\mu(2\hat i+\hat j+2\hat k)$.
15.
Find the angle between $\dfrac{x-2}{2}=\dfrac{y-1}{5}=\dfrac{z+3}{-3}$ and $\dfrac{x+2}{-1}=\dfrac{y-4}{8}=\dfrac{z-5}{4}$.
Answer Key
Section A — Multiple Choice Questions
- (B) $\vec b_1\cdot\vec b_2=0$
- (B) $\dfrac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|}$
- (B) $0$
- (B) parallel
- (B) $\vec b_1\times\vec b_2$
Section B — Short Answer (2 marks)
- $\cos^{-1}\!\left(\tfrac13\right)$.
- Perpendicular ($\vec b_1\cdot\vec b_2=0$).
- Yes.
- $\tfrac{1}{\sqrt2}$ (i.e. $45^\circ$).
Section C — Short Answer (3 marks)
- $\cos^{-1}\!\left(\tfrac89\right)$.
- $\cos^{-1}\!\left(\tfrac16\right)$.
- Parallel.
- $\lambda=8$.
Section D — Long Answer (5 marks)
- $\dfrac{3}{\sqrt2}$ (i.e. $\tfrac{3\sqrt2}{2}$) units.
- $\cos^{-1}\!\left(\dfrac{26}{9\sqrt{38}}\right)$.
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