Vidaara.orgClass 12 · Mathematics
CodeVID-M12-10-CRS-01
Vector (Cross) 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 cross product of two vectors is a:
- A.scalar
- B.vector
- C.number
- D.angle
2.
$\vec a\times\vec b=\vec 0$ means the vectors are:
- A.perpendicular
- B.parallel
- C.equal
- D.unit
3.
$|\vec a\times\vec b|$ represents the area of a:
- A.triangle
- B.parallelogram
- C.circle
- D.square only
4.
$\vec a\times\vec b$ compared with $\vec b\times\vec a$ is:
- A.equal
- B.its negative
- C.zero
- D.double
5.
$\vec a\times\vec a=$
- A.$|\vec a|^2$
- B.$\vec 0$
- C.$1$
- D.$\vec a$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find $(\hat i+\hat j)\times(\hat j+\hat k)$.
7.
Are $2\hat i+4\hat j$ and $\hat i+2\hat j$ parallel? (use cross product)
8.
Find $|\hat i\times\hat j|$.
9.
Find the area of the parallelogram with sides $\hat i+\hat j$ and $\hat j+\hat k$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find $\vec a\times\vec b$ for $\vec a=2\hat i+\hat j,\ \vec b=\hat i+2\hat j$.
11.
Find the area of the triangle with sides $\hat i+\hat j$ and $\hat j+\hat k$.
12.
Find a unit vector perpendicular to both $\hat i+\hat j$ and $\hat j+\hat k$.
13.
Find $|\vec a\times\vec b|$ if $|\vec a|=2,\ |\vec b|=3$ and the angle is $30^\circ$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Find the area of the triangle with vertices $A(1,1,1),\ B(1,2,3),\ C(2,3,1)$.
15.
If $\vec a=2\hat i-\hat j+\hat k,\ \vec b=\hat i+\hat j-2\hat k$, find $\vec a\times\vec b$ and a unit vector perpendicular to both.
Answer Key
Section A — Multiple Choice Questions
- (B) vector
- (B) parallel
- (B) parallelogram
- (B) its negative
- (B) $\vec 0$
Section B — Short Answer (2 marks)
- $\hat i-\hat j+\hat k$.
- Yes ($\vec a\times\vec b=\vec 0$).
- $1$.
- $\sqrt3$ sq units.
Section C — Short Answer (3 marks)
- $3\hat k$.
- $\tfrac{\sqrt3}{2}$ sq units.
- $\tfrac{1}{\sqrt3}(\hat i-\hat j+\hat k)$.
- $3$.
Section D — Long Answer (5 marks)
- $\tfrac{\sqrt{21}}{2}$ sq units.
- $\vec a\times\vec b=\hat i+5\hat j+3\hat k$; unit vector $\tfrac{1}{\sqrt{35}}(\hat i+5\hat j+3\hat k)$.
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