Vidaara.orgClass 12 · Mathematics
CodeVID-M12-10-DOT-01
Scalar (Dot) Product — 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.
The dot product of two vectors is a:
- A.vector
- B.scalar
- C.matrix
- D.unit vector
2.
$\vec a\perp\vec b$ if and only if $\vec a\cdot\vec b=$
- A.$1$
- B.$0$
- C.$|\vec a||\vec b|$
- D.$-1$
3.
$(\hat i+2\hat j+3\hat k)\cdot(2\hat i-\hat j+\hat k)=$
- A.$3$
- B.$5$
- C.$7$
- D.$0$
4.
$\vec a\cdot\vec a$ equals:
- A.$0$
- B.$|\vec a|$
- C.$|\vec a|^2$
- D.$1$
5.
The projection of $\vec a$ on $\vec b$ is:
- A.$\vec a\cdot\vec b$
- B.$\dfrac{\vec a\cdot\vec b}{|\vec b|}$
- C.$|\vec a||\vec b|$
- D.$\dfrac{\vec a}{|\vec b|}$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find $(\hat i+\hat j+\hat k)\cdot(\hat i-\hat j+\hat k)$.
7.
Are $\hat i+\hat j$ and $\hat i-\hat j$ perpendicular?
8.
Find $\vec a\cdot\vec a$ for $\vec a=2\hat i+\hat j+2\hat k$.
9.
Find the angle between $\hat i$ and $\hat j$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the angle between $\hat i+\hat j$ and $\hat i$.
11.
Find the projection of $2\hat i+3\hat j$ on $\hat i$.
12.
Find $\lambda$ if $\hat i+\lambda\hat j$ is perpendicular to $\hat i-\hat j$.
13.
Find $\vec a\cdot\vec b$ for $\vec a=\hat i+2\hat j+3\hat k,\ \vec b=2\hat i-\hat j+\hat k$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
If $|\vec a|=3,\ |\vec b|=4$ and $\vec a\cdot\vec b=6$, find the angle between $\vec a$ and $\vec b$.
15.
Find $\lambda$ for which $2\hat i+\lambda\hat j+\hat k$ and $\hat i-2\hat j+3\hat k$ are perpendicular.
Answer Key
Section A — Multiple Choice Questions
- (B) scalar
- (B) $0$
- (A) $3$
- (C) $|\vec a|^2$
- (B) $\dfrac{\vec a\cdot\vec b}{|\vec b|}$
Section B — Short Answer (2 marks)
- $1$.
- Yes.
- $9$.
- $90^\circ$.
Section C — Short Answer (3 marks)
- $45^\circ$.
- $2$.
- $\lambda=1$.
- $3$.
Section D — Long Answer (5 marks)
- $60^\circ$.
- $\lambda=\tfrac52$.
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