Explore topic-wise InterviewSolutions in .

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1.

Show that the vectors ` vec a=1/7(2 hat i+3 hat j+6 hat k), vec b=1/7(3 hat i-6 hat j+2 hat k), vec c=1/7(6 hat i+2 hat j-3 hat k)`are mutually perpendicular unit vectors.

Answer» Let the given vectors be `vec(a), vec(b), vec(c)` respectively .
Then, show that `|vec(a)|=|vec(b)|=|vec(c)|=1` and `vec(a).vec(b)=vec(b).vec(c)=vec(c).vec(a)=0`.
Discussion
2.

Let ` vec a , vec b ,a n d vec c`are vectors such that `| vec a|=3,| vec b|=4a n d| vec c|=5,a n d( vec a+ vec b)`is perpendicular to ` vec c ,( vec b+ vec c)`is perpendicular to ` vec aa n d( vec c+ vec a)`is perpendicular to ` vec bdot`Then find the value of `| vec a+ vec b+ vec c|`.

Answer» Let `vec(a), vec(b), vec(c)` be the given vectors such that
`{|vec(a)|=3, |vec(b)|=4, |vec(c)|=5}`, ...(i)
`{:(vec(a).(vec(b)+vec(c))=0),(vec(b).(vec(c)+vec(a))=0),(vec(c).(vec(a)+vec(b))=0):}}`. ...(ii)
`:. |vec(a)+vec(b)+vec(c)|^(2)=(vec(a)+vec(b)+vec(c)).(vec(a)+vec(b)+vec(c))`
`=vec(a).vec(a)+vec(a).(vec(b)+vec(c))+vec(b).(vec(c)+vec(a))+vec(b).vec(b)+vec(c).(vec(a)+vec(b))+vec(c).vec(c)`
`=|vec(a)|^(2)+|vec(b)|^(2)+|vec(c)|^(2)` [using (ii)]
`=(3^(2)+4^(2)+5^(2))=(9+16+25)=50`
Hence, `|vec(a)+vec(b)+vec(c)|=sqrt(50)=5sqrt(2)`.
Discussion
3.

Find the angle between two vectors ` vec aa n d vec b`with magnitudes `sqrt(3)`and `2`respectively and such that ` vec adot vec b=sqrt(6.)`

Answer» Let `theta` be the angle between `vec(a)` and `vec(b)`. Then,
`vec(a).vec(B)=sqrt(6) implies |vec(a)||vec(b)| cos theta = sqrt(6) implies (sqrt(3)) (2) cos theta =sqrt(6)`
`implies cos theta=sqrt(6)/(2sqrt(3))=1/sqrt(2) implies theta = cos^(-1) (1/sqrt(2))=pi/4`
Hence, the required angle is `pi/4`.
Discussion
4.

Dot products of a vector with vectors ` hat i- hat j+ hat k , 2 hat i+ hat j-3 hat k a n d hat i+ hat j+ hat k`are respectively 4, 0 and 2. Find the vector.

Answer» Let the required vector be `(x hat(i)+y hat(j)+z hat(k))`. Then,
`(x hat(i)+y hat(j)+z hat(k)). (hat(i)-hat(j)+hat(k))=4 implies x-y+z=4` ...(i)
`(x hat(i)+y hat(j)+z hat(k)). (2hat(i)+hat(j)-3hat(k))=0implies 2x+y-3z=0`. ...(ii)
`(x hat(i)+ y hat(j)+ z hat(k)).(hat(i)+hat(j)+hat(k))=2implies x+y+z=2` ...(iii)
On subtracting (i) from (iii) we get `2y=-2 implies y=-1`.
On adding (i) and (ii), we get `3x-2z=4` ...(iv)
On adding (i) and (iii), we get `2x+2z=6` ...(v)
On solving (iv) and (v), we get `x=2` and `z=1`.
`:. x=2, y=-1` and `z=1`
Hence, the required vector is `(2hat(i)-hat(j)+hat(k))`.
Discussion
5.

Express the vector `vec(a)=(6hat(i)-3hat(j)-6hat(k))` as sum of two vectors such that one is parallel to the vector `vec(B)=(hat(i)+hat(j)+hat(k))` and the other is perpendicular to `vec(b)`.

Answer» Correct Answer - `vec(a)=- (hat(i)+hat(j)+hat(k))+(7 hat(i)-2hat(j)-5hat(k))`
Let `vec(a)=lambda vec(b)+vec(c)`, where `vec(c) bot vec(b)`. Then, `vec(c)=(vec(a)-lambda vec(b)) bot vec(b)`.
`:. (vec(a)-lambda vec(b)). vec(b)=0 implies (vec(a).vec(b))-lambda (vec(b).vec(b))=0`
`implies (6-3-6)-lambda (1+1+1)=0 implies lambda = (-3)/3=-1`.
`:. vec(a)=-vec(b)+vec(c) implies vec(c)= (vec(a)+vec(b))`.
Discussion
6.

Let `vec(a)=(2hat(i)+3hat(j)+2 hat(k))` and `vec(b)=(hat(i)+2hat(j)+hat(k))`. Find the projection of (i) `vec(a)` on `vec(b)` and (ii) `vec(b)` on `vec(a)`.

Answer» Correct Answer - (i) `(5 sqrt(6))/3` (ii) `(10 sqrt(17))/17`
Discussion
7.

If ` vec a , vec b , vec c`are three vectors such that ` vec adot vec b= vec adot vec c`then show that ` vec a=0 or , vec b=c or vec a_|_( vec b- vec c)dot`

Answer» `vec(a).vec(b)=vec(a).vec(c) implies vec(a).vec(b)-vec(a).vec(c)=0implies vec(a).(vec(b)-vec(c))=0`
`implies vec(a)=vec(0)` or `(vec(b)-vec(c))=vec(0)` or `vec(a) bot (vec(b)-vec(c))`
`implies vec(a)=vec(0)` or `vec(b)=vec(c)` or `vec(a) bot (vec(b)-vec(c))`.
Hence, `vec(a).vec(b)=vec(a).vec(c)implies vec(a)=vec(0)` or `vec(b)=vec(c)` or `vec(a) bot (vec(b)-vec(c))`.
Discussion
8.

If `vec a=3 hat i- hat j` and `vec b=2 hat i+hat j-3 hat k`, then express `vec b` in the from `vec b=vec b_1+vec b_2`, where `vec b_1|| vec a` and `vec b_2 _|_ vec a`.

Answer» Correct Answer - `vec(B)=(vec(b_(1))+vec(b_(2)))`, where `vec(b_(1))=(3/2 hat(i)- 1/2 hat(j))` and `vec(b_(2))=(1/2 hat(i)+3/2 hat(j)-3 hat(k))`
Let `vec(b)=vec(b_(1))+vec(b_(2))`, where `vec(b_(1))||vec(a)` and `vec(b_(2)) bot vec(a)`.
Let `vec(b_(1))=lambda (3 hat(i)-hat(j))` and let `vec(b_(2))=x hat(i) + y hat(i) + z hat(k)`. Then,
`vec(b_(2)) bot vec(a) implies (x hat(i)+ y hat(j)+z hat(k)). (3 hat(i)-hat(j))=0 implies 3x-y =0`.
`:. vec(b)=(vec(b)_(1)+vec(b)_(2)) implies (2 hat(i)+hat(j)-3hat(k))=lambda (3hat(i)-hat(j))+(x hat(i)+y hat(j)+ z hat(k))`
`implies (2 hat(i)+hat(j)-3hat(k))=(3 lambda + x) hat(i)+(y-lambda) hat(j)+z hat(k)`
`implies z=-3, y- lambda =1` and `3 lambda +x=2`
`implies z=-3, lambda =(y-1)` and `3(y-1)+x=2`
`implies z=-3, lambda =(y-1)` and `3y+x=5`.
On solving `3x-y=0` and `x+3y=5`, we get `x=1/2` and `y=3/2, z=-3` and `lambda =1/2`.
`:. vec(b_(1))=1/2 (3hat(i)-hat(j))` and `vec(b_(2))=1/2 hat(i) +3/2 hat(j) - 3hat(k)`.
Discussion
9.

If ` vec a`and ` vec b`aretwo non-collinear unit vectors such that `| vec a+ vec b|=sqrt(3),`find `(2 vec a-5 vec b)dot(3 vec a+ vec b)`.

Answer» Correct Answer - `(-11)/2`
Given : `|vec(a)|=1, |vec(b)|=1` ad `|vec(a)+vec(b)|=sqrt(3)`.
`:. |vec(a)+vec(b)|^(2)=(vec(a)+vec(b)).(vec(a)+vec(b))`
`=(vec(a).vec(a))+2(vec(a).vec(b))+(vec(b).vec(b))`
`= |vec(a)|^(2)+ 2(vec(a).vec(b))+|vec(b)|^(2)=2+2(vec(a).vec(b))`.
`:. 2[1+(vec(a).vec(b))]=3 implies (vec(a).vec(b))=1/2`.
Now, find `(2 vec(a)- 5 vec(b)).(3 vec(a)+vec(b))`.
Discussion