InterviewSolution
Saved Bookmarks
InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
When does `|vec(a)+vec(b)|=|vec(a)|+|vec(b)|` hold? |
|
Answer» Correct Answer - `vec(a) and vec(b)` are parallel or collinear `|vec(a)+vec(b)|=|vec(a)|+|vec(b)|rArr|vec(a)+vec(b)|={|vec(a)|+|vec(b)|}^(2)` `rArr (vec(a)+vec(b)).(vec(a)+vec(b))=|vec(a)|+|vec(b)|+2|vec(a)||vec(b)|` `rArrvec(a).vec(b)+2vec(a).vec(b)+vec(b).vec(b)=|vec(a)|^(2)+|vec(b)|^(2)+2|vec(a)||vec(b)|` `rArr|vec(a)|^(2)+2vec(a).vec(b)+|vec(b)|^(2)=|vec(a)|^(2)+|vec(b)|+2|vec(a)||vec(b)|` `rArr 2(vec(a).vec(b))=2(|vec(a)||vec(b)|)rArr vec(a).vec(b)=|vec(a)||vec(b)|` `rArr |vec(a)||vec(b)|cos theta=|vec(a)||vec(b)|rArr cos theta =1 rArr theta=0^(@)`. `:.` either `vec(a) and vec(b)` are parallel or collinear. |
|
| 2. |
What is the angle which the vector `(hat(i)+hat(j)+sqrt(2)hat(k))` makes with the z-axis ?A. `(pi)/(4)`B. `(pi)/(3)`C. `(pi)/(6)`D. `(2pi)/(3)` |
|
Answer» Correct Answer - A `vec(a)=(hat(i)+hat(j)+sqrt(2)hat(k))rArr|vec(a)|=sqrt(1^(2)+1^(2)+(sqrt(2))^(2))=sqrt(4)=2`. Direction ratios of `vec(a)` are 1, 1, `sqrt(2)`. Direction cosines of `vec(a)` are `(1)/(2), (1)/(2), (sqrt(2))/(2), i.e., (1)/(2), (1)/(2), (1)/(sqrt(2))`. `:. cos lambda=(1)/(sqrt(2))rArr lambda=(pi)/(4)`. |
|
| 3. |
If `vec(a)` and `vec(b)` are vectors such that `|vec(a)|=sqrt(3), |vec(b)|=2` and `vec(a).vec(b)=sqrt(6)` then the angle between `vec(a) and vec(b)` isA. `(pi)/(6)`B. `(pi)/(3)`C. `(pi)/(4)`D. `(2pi)/(3)` |
|
Answer» Correct Answer - C `cos theta=(vec(a).vec(b))/(|vec(a)||vec(b)|)=(sqrt(6))/(sqrt(3)xx2)=(1)/(sqrt(2))rArr theta=(pi)/(4)`. |
|
| 4. |
Find the direction cosines of a vector which is equally inclined to the x-axis, y-axis and z-axis. |
|
Answer» Correct Answer - `(1)/(sqrt(3)), (1)/(sqrt(3)), (1)/(sqrt(3))` `cos^(2)alpha+cos^(2)alpha+cos^(2)alpha=1rArr3cos^(2)alpha=1rArrcos alpha=(1)/(sqrt(3))`. So, the direction cosines of the vector are `(1)/(sqrt(3)), (1)/(sqrt(3)),(1)/(sqrt(3))`. |
|
| 5. |
Find the direction cosines of the vector ` hat i+2 hat j+3 hat k`. |
|
Answer» Correct Answer - `(1)/(sqrt(14)), (2)/(sqrt(14)),(3)/(sqrt(14))` Direction ratios of `vec(a)` are 1,2,3. And, `sqrt(1^(2)+2^(2)+3^(2))=sqrt(14)`. `:.` direction cosines of `vec(a)` are `(1)/(sqrt(14)),(2)/(sqrt(14)),(3)/(sqrt(14))`. |
|
| 6. |
If ` vec a=x hat i+2 hat j- z hat k`and ` vec b=3 hat i- y hat j+ hat k`are two equal vectors, thenwrite the value of `x+y+zdot` |
|
Answer» `vec(a)=vec(b)rArr(xhat(i)+2hat(j)-zhat(k))-(3hat(i)-yhat(j)+hat(k))` `rArr x=3, -y=2 and -z=1 rArr x=3, y=-2 and z=-1` `rArr (x+y+z)=[3+(-2)+(-1)]=0`. |
|
| 7. |
If `vec(a)` and `vec(B)` are two vectors such that `|vec(a)|=|vec(b)|=sqrt(2)` and `vec(a).vec(b)=-1`, find the angle between `vec(a)` and `vec(b)`.A. `(pi)/(6)`B. `(pi)/(4)`C. `(pi)/(3)`D. `(2pi)/(3)` |
|
Answer» Correct Answer - D `cos theta= (vec(a).vec(b))/(|vec(a)||vec(b)|)=(-1)/(sqrt(2)xxsqrt(2))=(-1)/(2)rArrtheta=(2pi)/(3)`. |
|
| 8. |
If `vec(a)=(hat(i)+2hat(j)-3hat(k)) and vec(b)=(3hat(i)-hat(j)+2hat(k))` then the angle between `(vec(a)+vec(b))` and `(vec(a)-vec(b))` isA. `(pi)/(3)`B. `(pi)/(4)`C. `(pi)/(2)`D. `(2pi)/(3)` |
|
Answer» Correct Answer - C `(vec(a)+vec(b))=(4hat(i)+hat(j)-hat(k)) and (vec(a)-vec(b))=(-2hat(i)+3hat(j)-5hat(k))`. `:. cos theta=((vec(a)+vec(b)).(vec(a).vec(b)))/(|vec(a)+vec(b)|.|vec(a)-vec(b)|)=(-8+3+5)/(sqrt(18)xxsqrt(38))=0rArrtheta=(pi)/(2)` |
|
| 9. |
If `hat(a) and hat(b)` are unit vectors such that `(hat(a) + hat(b))` is a unit vector, what is the angle between `hat(a) and hat(b)`? |
|
Answer» Correct Answer - `(2pi)/(3)` `(hat(a)+hat(b)).(hat(a)+hat(b))=1rArra^(2)+b^(2)+2hat(a).hat(b)=1` `rArr1+1+2hat(a).hat(b)=1rArr2hat(a).hat(b)=-1` `rArrhat(a).hat(b)=(-1)/(2)rArr|hat(a)||hat(b)|cos theta=(-1)/(2)` `rArr cos theta = (-1)/(2)rArr theta=(2pi)/(3)`. |
|
| 10. |
Write the direction cosines of the vector `-2 hat i+ hat j-5 hat k`A. `-2, 1, -5`B. `(1)/(3),(-1)/(6),(-5)/(6)`C. `(2)/(sqrt(30)), (1)/(sqrt(30)),(5)/(sqrt(30))`D. `(-2)/(sqrt(30)),(1)/(sqrt(30)),(-5)/(sqrt(30))` |
|
Answer» Correct Answer - D `|vec(a)|=sqrt((-2)^(2)+1^(2)+(-5)^(2))=sqrt(3)`. `:.` direction cosines of `vec(a)` are `(-2)/(sqrt(30)),(1)/(sqrt(30)), (-5)/(sqrt(30))`. |
|
| 11. |
Find a unit vector in the direction of ` vec a=2 hat i- 3 hat j+6 hat k`A. `(hat(i)-(3)/(2)hat(j)+3hat(k))`B. `((2)/(5)hat(i)-(3)/(5)hat(j)+(6)/(5)hat(k))`C. `((2)/(7)hat(i)-(3)/(7)hat(j)+(6)/(7)hat(k))`D. none of these |
|
Answer» Correct Answer - C `|vec(a)|=sqrt(2^(2)+(-3)^(2)+6^(2))=sqrt(49)=7`. `:.` required unit vector is `((2)/(7)hat(i)-(3)/(7)hat(j)+(6)/(7)hat(k))`. |
|
| 12. |
Find the area of a parallelogram whose adjacent sides are given by the vectors ` -> a=3 hat i+ hat j+4 hat k`and ` -> b= hat i- hat j+ hat k`.A. `sqrt(42)` sq unitsB. 6 sq unitsC. `sqrt(35)`sq unitsD. none of these |
|
Answer» Correct Answer - A `(vec(a)xxvec(b))=|(hat(i),hat(j),hat(k)),(3,1,4),(1,-1,1)|=(1+4)hat(i)-(3-4)hat(j)+(-3-1)hat(k)=5hat(i)+hat(j)-4hat(k)`. Required area `=|vec(a)xxvec(b)|=sqrt(25+1+16)=sqrt(42)`sq units. |
|
| 13. |
If ` vec a , vec b , vec c`are three mutually perpendicular unit vectors,then prove that `| vec a+ vec b+ vec c|=sqrt(3)`A. 1B. `sqrt(2)`C. `sqrt(3)`D. 2 |
|
Answer» Correct Answer - C Given `a^(2)=b^(2)=c^(2)=1 and vec(a).vec(b)=vec(b).vec(c)=vec(c).vec(a)=0`. `:. |vec(a)+vec(b)+vec(c)|^(2)=(vec(a)+vec(b)+vec(c)).(vec(a)+vec(b)+vec(c))` `=vec(a).vec(a)+vec(b).vec(b)+vec(c).vec(c)+2(vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a))` `=a^(2)+b^(2)+c^(2)+2xx0=(1+1+1+0)=3`. Hence, `|vec(a)+vec(b)+vec(c)|=sqrt(3)`. |
|
| 14. |
Write the projectionof the vector ` hat i- hat j`on the vector ` hat i+ hat j` |
|
Answer» Required projection `=((hat(i)-hat(j)).(hat(i)+hat(j)))/(|hat(i)+hat(j)|)=((hat(i).hat(i)+hat(i).hat(j)-hat(j).hat(i)-hat(j)-hat(j)))/(sqrt(1^(2)+1^(2)))` `=((1+0-0-1))/(sqrt(2))=0`. |
|
| 15. |
The unit vector normal to the plane containing `vec(a)=(hat(i)-hat(j)-hat(k)) and vec(b)=(hat(i)+hat(j)+hat(k)) ` isA. `(hat(j)-hat(k))`B. `(-hat(j)+hat(k))`C. `(1)/(sqrt(2))(-hat(j)+hat(k))`D. `(1)/(sqrt(2))(-hat(i)+hat(k))` |
|
Answer» Correct Answer - C `(vec(a)xxvec(b))=|(hat(i),hat(j),hat(k)),(1,-1,-1),(1,1,1)|=(-2hat(j)+2hat(k))` `rArr |vec(a)xxvec(b)|=sqrt((-2)^(2)+2^(2))=sqrt(8)=2sqrt(2)` `rArr ` required vector`=(2(-hat(j)+hat(k)))/(2sqrt(2))=(1)/(sqrt(2))(-hat(j)+hat(k))`. |
|
| 16. |
Prove that (i) `[hat(i)hat(j)hat(k)]=[hat(j)hat(k)hat(i)]=[hat(k)hat(j)hat(i)]=1` (ii)`[hat(i)hat(k)hat(j)]=[hat(k)hat(j)hat(i)]=[hat(j)hat(i)hat(k)]=1` |
|
Answer» Correct Answer - B `[hat(i)hat(j)hat(k)]=(hat(i)xxhat(j)).hat(k)=hat(k).hat(k)=1`. |
|
| 17. |
Find the area aparallelogram whose diagonals are ` vec a=3 hat i+ hat j-2 hat ka n d vec b= hat i-3 hat j+4 hat kdot`A. `7sqrt(3)` sq unitsB. `5sqrt(3)` sq unitsC. `3sqrt(5)` sq unitsD. none of these |
|
Answer» Correct Answer - B `(vec(d_(1))xxvec(d_(2)))=|(hat(i),hat(j),hat(k)),(3,1,-2),(1,-3,4)|=(4-6)hat(i)-(12+2)hat(j)+(-9-1)hat(k)` `=(-2hat(i)-14hat(j)-10hat(k)) ` `rArr|vec(d_(1))xxvec(d_(2))|=sqrt((-2)^(2)(-14)^(2)+(-10)^(2))=sqrt(300)=10sqrt(3)` `rArr` area of the ||gm `=(1)/(2)|vec(d_(1))xxvec(d_(2))|=5sqrt(3)` sq units. |
|
| 18. |
Find the angle between thevectors ` hat i-2 hat j+3 hat ka n d3 hat i-2 hat j+ hat kdot`A. `cos^(-1).(5)/(7)`B. `cos^(-1).(3)/(5)`C. `cos^(-1).(3)/(sqrt(14))`D. none of these |
|
Answer» Correct Answer - A `cos theta=(vec(a).vec(b))/(|vec(a)||vec(b)|)=([1xx3+(-2)xx(-2)+3xx1])/([sqrt(1^(2)+(-2)^(2)+3^(2))][sqrt(3^(2)+(-2)^(2)+1^(2))])=(10)/(14)=(5)/(7)rArr theta=cos^(-1).(5)/(7)`. |
|
| 19. |
Write a unit vector in the direction of the sum of the vectors ` vec a=2 hat i+2 hat j-5 hat k a n d vec b=2 hat i+ hat j-7 hat kdot` |
|
Answer» Correct Answer - `(1)/(13)(4hat(i)+3hat(j)-12hat(k))` Required unit vector`=((vec(a)xxvec(b)))/(|vec(a)xxvec(b)|)=((4hat(i)+3hat(j)-12hat(k)))/(sqrt(16+9+144))=(1)/(13)(4hat(i)+3hat(j)-12hat(k))`. |
|
| 20. |
Find `| vec axx vec b|`, if ` vec a= hat i-7 hat j+7 hat ka n d vec b=3 hat i-2 hat j+2 hat kdot` |
|
Answer» Correct Answer - `19sqrt(2)` `(vec(a)xxvec(b))=|(hat(i),hat(j),hat(k)),(1,-7,7),(3,-2,2)|=(-14+14)hat(i)-(2-21)hat(k)` `=(19hat(j)+19hat(k))` `:. |vec(a)xxvec(b)|=sqrt(0^(2)+(19)^(2)+(19)^(2))=sqrt(2xx(19)^(2))=19sqrt(2)`. |
|
| 21. |
If ` vec a= hat i+ hat j+ hat k , vec b=4 hat i-2 hat j+3 hat k`and ` vec c= hat i-2 hat j+ hat k`, find avector of magnitude 6 units which is parallel to the vector `2 vec a- vec b+3 vec cdot` |
|
Answer» Correct Answer - `(2hat(i)-4hat(j)+4hat(k))` `(2vec(a)-vec(b)+3vec(c))=(2hat(i)+2hat(j)+2hat(k))-(4hat(i)-2hat(j)+3hat(k))+(3hat(i)-6hat(j)+3hat(k))` `=(hat(i)-2hat(j)+2hat(k))`. `:.` required vector`=(6(hat(i)-2hat(j)+2hat(k)))/(sqrt(a^(2)(-2)^(2)+2^(2)))=(6(hat(i)-2hat(j)+2hat(k)))/(3)=(2hat(i)-4hat(j)+4hat(k))` |
|
| 22. |
Write a vector of magnitude 15 units in the direction of vecor `(hat(i)-2hat(j)+2hat(k))`. |
|
Answer» Correct Answer - `(5hat(i)-10hat(j)+10hat(k))` Required vector`=(15(hat(i)-2hat(j)+2hat(k)))/(sqrt(1^(2)(-2)^(2)+2^(2)))=5(hat(i)-2hat(j)+2hat(k))=(5hat(i)-10hat(j)+10hat(k))`. |
|
| 23. |
Find `lambda,`when the projection of ` vec a=lambda hat i+ hat j+4 hat k on vec b=2 hat i+6 hat j+3 hat k i s 4 `units. |
|
Answer» Correct Answer - `lambda=5` Clearly, `(vec(a).vec(b))/(|vec(b)|)=4rArr ((2lambda+6+12))/(sqrt(4+36+9))=4rArr (2lambda+18)=28 rArr 2lambda=10 rArr lambda=5`. |
|
| 24. |
Find the volume ofthe parallelepiped whose coterminous edges are represented by the vector: ` vec a=2 hat i+3 hat j+4 hat k , vec b= hat i+2 hat j- hat k , vec c=3 hat i- hat j+2 hat k`.A. 21 cubic unitsB. 14 cubic unitsC. 7 cubic unitsD. none of these |
|
Answer» Correct Answer - C We have `(vec(a)xxvec(b)).vec(c)=|(2,-3,4),(1,2,-1),(3,-1,2)|=2(4-1)+3(2+3)+4(-1-6)` `=(6+15-28)=-7`. `:.` volume of the parallelopiped `=|(vec(a)xxvec(b)).vec(c)|=|-7|=7` cubic units. |
|
| 25. |
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors `vec(a)=2hat(i)-3hat(j)+hat(k),vec(b)=hat(i)-hat(j)+2hat(k)` and `vec(c)=2hat(i)+hat(j)-hat(k)`. |
|
Answer» `[vec(a)vec(b)vec(c)]=|(2,-3,1),(1, -1,2),(2,1,-1)|=|(0,-1,-3),(1,-1,2),(0,3,-5)|" "[R_(1)rarrR_(1)-2R_(2),andR_(3)rarrR_(3)-2R_(2)]` `=(-1).(5+9)=-14`. `:.` volume of the parallelepiped `=|[vec(a)vec(b)vec(c)]|=|-14|=14` cubic units. |
|
| 26. |
Writhe the projection of the vector `7 hat i+ hat j-4 hat k`on the vector `2 hat i+6 hat j+3 hat kdot` |
|
Answer» Correct Answer - 1 Required projection `=((hat(i)+hat(j)+hat(k)).hat(j))/(|hat(j)|)=((0+1+0))/(1)=1`. |
|
| 27. |
Find the sum of the vectors `vec(a)=(hat(i)-3hat(k)), vec(b)=(2hat(j)-hat(k)) and vec(c)=(2hat(i)-3hat(j)+2hat(k))`. |
|
Answer» Correct Answer - `(3hat(i)-hat(j)-2hat(k))` `(vec(a)+vec(b)+vec(c))=(1+0+2)hat(i)+(0+2-3)hat(j)+[(-3)+(-1)+2]hat(k)` `=(3hat(i)-hat(j)-2hat(k))`. |
|
| 28. |
Show that the four points A, B, C and D with position vectors`4 hat i+5 hat j+ hat k ,-( hat j+ hat k),3 hat j+9 hat j+4 hat k a n d 4( hat i+ hat j+ hat k)`, respectively are coplanar. |
|
Answer» Let the given points be A, B, C, D respectively. Point a, B, C, D are coplanar `hArr vec(AB), vec(AC)and vec(AD)` are coplanar `hArr [vec(AB)vec(AC)vec(AD)]=0`. Now, `vec(AB)=("p.v.of B")-("p.v. of A")` `=(-hat(j)-hat(k))-(4hat(i)+5hat(j)+hat(k))=(-4hat(i)-6hat(j)-2hat(k))` `vec(AC)=("p.v of C")-("p.v. of A")` `=(3 hat(i)+9hat(j)+4hat(k))-(4hat(i)+5hat(j)+hat(k))=(-hat(i)+4hat(j)+3hat(k))` `vec(AD)=("p.v. of D")-("p.v. of A")` `=(-4hat(i)+4hat(j)+4hat(k))-(4hat(i)+5hat(j)+hat(k))=(-8hat(i)-hat(j)+3hat(k))`. `:. [vec(AB)vec(AC)vec(AD)]=|(-4,-6,-2),(-1 ,4,3),(-8,-1,3)|` `=|(0,-22, -14),(-1, 4, 3),(0, -21, -33)|{{:(R_(1)rarrR_(1)-4R_(2)),(R_(3)rarr R_(3)-8R_(2)):}}` `=-(-1)[462-462]=0`. `:. vec(AB), vec(AC) and vec(AD)` are coplanar. Hence, the points A, B, C, D are coplanar. |
|
| 29. |
Find a vector in the direction of vector `2 hat i-3 hat j+6 hat k`which has magnitude 21 units. |
|
Answer» Correct Answer - `(6hat(i)-9hat(j)+18hat(k))` Required vector `=(21(2hat(i)-3hat(j)+6hat(k)))/(sqrt(2^(2)+(-3)^(2)+6^(2)))=(21(2hat(i)-3hat(j)+6hat(k)))/(sqrt(49))` `=3(2hat(i)-3hat(j)+6hat(k))=(6hat(i)-9hat(j)+18hat(k))`. |
|
| 30. |
If `vec(a)=(hat(i)-2hat(j)+3hat(k)) and vec(b)=(hat(i)-3hat(k))` then `|vec(b)xx 2vec(a)|=?`A. `10sqrt(3)`B. `5sqrt(17)`C. `4sqrt(19)`D. `2sqrt(23)` |
|
Answer» Correct Answer - C `vec(b)=(hat(i)+0hat(j)-3hat(k))and 2hat(a)=(2hat(i)-4hat(j)+6hat(k))`. `:. (vec(b)xx2vec(a))=|(hat(i),hat(j) ,hat(k)),(1,0,-3),(2,-4,6)|=(0-12)hat(i)-(6+6)hat(j)+(-4-0)hat(k)` `=(-12hat(i)-12hat(j)-4hat(k))`. `:. |vec(b)xx2vec(a)|^(2)={(-12)^(2)+(-12)^(2)+(-4)^(2)}=(144+144+16)=304`. Hence, `|vec(b)xx2vec(a)|=sqrt(304)=4sqrt(19)`. |
|
| 31. |
Find the value of `lambda` for which the four points with position vectors `(hat(i)+2hat(j)+3hat(k)), (3hat(i)-hat(j)+2hat(k)), (-2hat(i)+lambda hat(j)+hat(k))and (6hat(i)-4hat(j)+2hat(k))` are coplanar. |
| Answer» Correct Answer - `lambda=3` | |
| 32. |
The volume of the parallelepiped whose edges are `(-12hat(i)+lambdahat(k)), (3hat(j)-hat(k))and (2hat(i)+hat(j)-15hat(k))` is 546 cubic units. Find the value of `lambda`. |
| Answer» Correct Answer - `lambda=-3` | |
| 33. |
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors (i) `vec(a)=hat(i)+hat(j)+hat(k), vec(b)=hat(i)-hat(j)+hat(k), vec(c)=hat(i)+2hat(j)-hat(k)` (ii) `vec(a)=-3hat(i)+7hat(j)+5hat(k), vec(b)=-5hat(i)+7hat(j)-3hat(k), vec(c)= 7 hat(i)-5hat(j)-3hat(k)` (iii)`vec(a)=hat(i)-2hat(j)+3hat(k), vec(b)=2hat(i)+hat(j)-hat(k), vec(c)=hat(j)+hat(k)` (iv) `vec(a)=6hat(i), vec(b)=2hat(j), vec(c)=5hat(i)` |
| Answer» Correct Answer - (i) 4 cubic units (ii) 264 cubic units (iii) 12 cubic units (iv) 60 cubic units | |
| 34. |
If `vec(a)=(2hat(i)-hat(j)+hat(k)), vec(b)=(hat(i)-3hat(j)-5hat(k)) and vec(c)=(3hat(i)-4hat(j)-hat(k))`, find `[vec(a)vec(b)vec(c)]` and interpret the result. |
| Answer» Correct Answer - 0, the given vectors are coplanar | |
| 35. |
If the vectors `vec(a)=3hat(i)+hat(j)-2hatk and vec(b)=hat(i)+lambda hat(j)-3hat(k)` are perpendicular to each other then `lambda=?`A. -3B. -6C. -9D. -1 |
|
Answer» Correct Answer - C `vec(a)_|_vec(b)rArrvec(a).vec(b)=0` `rArr3xx1+1xxlambda+(-2)xx(-3)=0rArr lambda=-9` |
|
| 36. |
If `vec(a)=(2hat(i)+4hat(j)-k^(2)) and vec(b)=(3hat(i)-2hat(j)+lambda hat(k))` be such that `vec(a) _|_ vec(b)` then `lambda=`?A. 2B. -2C. 3D. -3 |
|
Answer» Correct Answer - B `vec(a)_|_vec(b)rArrvec(a).vec(b)=0rArr2xx3+4xx(-2)+(-1)xxlambda=(0)` `rArr lambda=(6-8)=-2`. |
|
| 37. |
Find the value of `lambda` so that the vectors `vec(a)=2hat(i)-3hat(j)+hat(k),vec(b)=hat(i)+2hat(j)-3hat(k) a nd vec(c)=hat(j)+lambda hat(k)` are coplanar. |
|
Answer» The given vectors will be coplanar if `[vec(a)vec(b)vec(c)]=0`. Now, ` [vec(a)vec(b)vec(c)]=0hArr|(2,-3,1),(1,2,-3),(0,1,lambda)|=0hArr|(0,-7,7),(1,2,-3),(0,1,lambda)|=0 [R_(1)rarrR_(1)-2R_(2)]hArr(-1)(-7lambda-7)=0hArr7lambda + 7 = 0 hArr lambda=-1`. Hence, the given vectors are coplanar when `lambda=-1`. |
|
| 38. |
Find `[vec(a)vec(b)vec(c)]`, when `(i) vec(a)=2hat(i)+hat(j)+3hat(k), vec(b)=-hat(i)+2hat(j)+hat(k) and vec(c)=3hat(i)+hat(j)+2hat(k)` (ii) `vec(a)=2hat(i)-3hat(j)+4hat(k), vec(b)=hat(i)+2hat(j)-hat(k) and vec(c)=3hat(i)-hat(j)+2hat(k)` (iii) `vec(a) = 2 hat(i)-3hat(j), vec(b)=hat(i)+hat(j)-hat(k) and vec(c)=3hat(i)-hat(k)` |
|
Answer» Correct Answer - -10 `(vec(a)xxvec(b))=|(hat(i), hat(j),hat(k)),(-1,2,1),(3,1,2)|=(3hat(i)+5hat(j)-7hat(k))` `:. vec(a).(vec(b)xxvec(c))=(2hat(i)+hat(j)+3hat(k)).(3hat(i)+5hat(j)-7hat(k))=(6+5-21)=-10`. |
|
| 39. |
Which of the following is meaningless?A. `vec(a).(vec(b)xxvec(c))`B. `vec(a)xx(vec(b).vec (c))`C. `(vec(a)xxvec(b)).vec(c)`D. none of these |
|
Answer» Correct Answer - B Clearly, `vec(a) xx (vec(b).vec(c))` is meaningless. |
|
| 40. |
It is given that the vectors`vec(a)=(2hat(i)-2hat(k)), vec(b)=hat(i)+(lambda +1)hat(j) and vec(c)=(4hat(j)+2hat(k))` are coplanar. Then, the value of `lambda` isA. `(1)/(2)`B. `(1)/(3)`C. 2D. 1 |
|
Answer» Correct Answer - D Since `vec(a),vec(b),vec(c)` are coplanar, we must have `[vec(a)vec(b)vec(c)]=0` Now, `[vec(a)vec(b)vec(c)]=|(2,0,-2),(1,lambda+1,0),(0,4,2)|` `=2(2lambda+2-0)-2(4-0)` `=4lambda+4-8=4lambda-4` `:. [vec(a)vec(b)vec(c)]=0 hArr 4lambda-4=0hArr 4 lambda=4 hArr lambda=1`. |
|
| 41. |
Find the value of `lambda` for which the vectors `vec(a), vec(b), vec(c)` are coplanar, where (i) `vec(a)=(2hat(i)-hat(j)+hat(k)), vec(b) = (hat(i)+2hat(j)+3hat(k) ) and vec(c)=(3 hat(i)+lambda hat(j) + 5 hat (k))` (ii) `vec(a)lambda hat(i)-10 hat(j)-5k^(2), vec(b) =-7hat(i)-5hat(j) and vec(c)= hat(i)--4hat(j)-3hat(k)` (iii) `vec(a)=hat(i)-hat(j)+hat(k), vec(b)= 2hat( i) + hat(j)-hat(k) and vec(c)= lambda hat(i) - hat(j) + lambda hat(k)` |
| Answer» Correct Answer - (i)`lambda=-4` (ii) `lambda=-3` (iii) `lambda=1` | |
| 42. |
Show that the vectors` vec a=-2 hat i-2 hat j+4 hat k , vec b=-2 hat i+4 hat j-2 hat k a n d vec c=4 hat i-2 hat j-2 hat k`are coplanar. |
|
Answer» `[vec(a)vec(b)vec(c)]=|(-2,-2,4),(-2,4,-2),(4,-2,-2)|=(-2)(-8-4)+2(4+8)+4(4-16)` `=(24+24-48)=0`. Hence `vec(a), vec(b),vec(c)` are coplanar. |
|
| 43. |
Find `lambda`for which the points `A(3, 2, 1), B(4, lambda, 5), C(4, 2, -2) a n d D(6, 5, -1)`are coplanar. |
| Answer» Correct Answer - `lambda=5` | |
| 44. |
Write the value of `p`for which ` vec a=3 hat i+2 hat j+9 hat k a n d vec b= hat i+p hat j+3 hat k`are parallel vectors. |
|
Answer» Correct Answer - `lambda=(2)/(3)` Clearly ,`vec(a)||vec(b) hArr vec(a) xxvec(b)=vec(0)`. `(vec(a)xxvec(b))=|(hat(i),hat(j),hat(k)),(1,lambda,3),(3,2,9)|=(9lambda-6)hat(i)-0hat(j)+(2-3lambda)hat(k)` `:. (vec(a)xx vec(a)xxvec(b))=vec(0) hArr 9 lambda-6=0 and 2-3lambda=0 hArr lambda = (2)/(3)`. |
|
| 45. |
Giventhat ` -> adot -> b=0`and ` -> axx -> b= ->0`.What can you conclude about the vectors ` -> a`and ` -> b`. |
|
Answer» Correct Answer - `vec(a)=vec(0) or vec(b)=vec(0)` `(vec(a)xxvec(b)=vec(0) and vec(a).vec(b)=0)` only when `vec(a)=vec(0) or vec(b)=vec(0)`. |
|