Home
Class 12
MATHS
Let vec a , vec b ,a n d vec c be non-c...

Let ` vec a , vec b ,a n d vec c` be non-coplanar unit vectors, equally inclined to one another at an angle`theta` . If ` vec axx vec b+ vec bxx vec c=p vec a+q vec b+r vec c ,` find scalars `p ,qa n dr` in terms of `thetadot`

Text Solution

Verified by Experts

The correct Answer is:
`p=r= (1)/(sqrt(1+2 cos 0) , q= (-2cos 0)/(sqrt(1+2 cos 0)`
Since `vec(a) ,vec(b), vec(c )` are non- coplanar vectors .
`rArr [vec(a) vec(b) vec( c)] ne 0`
Also `vec(a) xx vec(b) + vec(b) xx vec( c) = p vec(a) + q vec(b) + r vec(c )`
Taking dot product with `vec(a) , vec(b) " and " vec( c)` respectively both sides we get
`p+q cos theta + r cos theta = [vec(a) vec(b) vec( c)] " "......(i)`
`p cos theta + q r cos theta =0" " .....(ii)`
and `p cos 0 + q cos theta + r = [ vec(a) vec( b) vec(c )] " " .....(ii)`
On adding above equations
`p+q + r= (2 [vec(a) vec(b) vec(c )])/(2 cos theta +1) " ".....(iii)`
On multiplying Eq. (iv) by cos `theta` and subtracting Eq. (i) we get
` p (cos theta -1) = (2[vec(a) vec(b)vec(c )])/(2 cos theta +1) -[vec(a) vec( b) vec( c)]`
`rArr p = ([vec(a) vec(b ) vec( c)])/((1-cos theta )(1-cos theta))`
Similarly `q= (-2[vec(a) vec(b) vec( c)] cos theta)/((1+2 cos theta )(1 - cos theta))`
and `r= ([vec(a)vec(b) vec(c )])/((1+2 cos theta) (1-cos theta))`
Now `[vec(a) vec(b )vec( c)]^(2)= |{:(vec(a)"."vec(a),,vec(a)"."vec(b),,vec(a)"."vec(b)),(vec(b)"."vec(a),,vec(b)"."vec(b),,vec(b)"."vec(c)),(vec(c)"."vec(a),,vec(c) "."vec(b),,vec(c)"."vec(c)):}|= |{:(1,,cos theta,,cos theta),(cos theta,,1,,cos theta),(cos theta,,cos theta,,1):}|`
Applying `R_(1) to R_(1) + R_(2) +R_(2)`
`=(1+2 cos theta ) |{:(1,,1,,1),(cos theta,,1,,cos theta),(cos theta ,,cos theta ,,1):}|` ltbr. Applying `C_(2) to C_(2) -C_(1) ,C_(3) to C_(3) -C_(1)`
`=(1+ 2 cos theta ) .|{:(1,,0,,0),(cos theta ,,1-cos theta ,,0),(cos theta ,,0 ,,1-cos theta):}|`
` =(1+2 cos theta) . (1-cos theta)^(2)`
`rArr [vec(a) vec(b) vec(c )] = (sqrt(1+2 cos theta ).(1-cos theta )`
`:. P = (1)/(sqrt(1+2 cos theta) ) , q = (-2 cos theta)/(sqrt(1+2cos theta)) " and " r= (1)/(sqrt(1 +2cos theta))`
Promotional Banner

Similar Questions

Explore conceptually related problems

If vectors vec a , vec b ,a n d vec c are coplanar, show that | vec a vec b vec c vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec b vec bdot vec c|=0

If vec a , vec b ,a n d vec c are three non-coplanar non-zero vecrtors, then prove that ( vec a . vec a) vec bxx vec c+( vec a . vec b) vec cxx vec a+( vec a . vec c) vec axx vec b=[ vec b vec c vec a] vec a

If vec a , vec b and vec c are non-coplanar vectors, prove that vectors 3vec a-7 vec b-4 vec c ,3 vec a -2 vec b+ vec c and vec a + vec b +2 vec c are coplanar.

Let vec a* vec b=0,w h e r e vec aa n d vec b are unit vectors and the unit vector vec c is inclined at an angle theta to both vec aa n d vec bdot If vec c=m vec a+n vec b+p( vec axx vec b),(m ,n , p in R), then a. -pi/4lt=thetalt=pi/4 b. pi/4lt=thetalt=(3pi)/4 c. 0lt=thetalt=pi/4 d. 0lt=thetalt=(3pi)/4

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

If vec a , vec b , vec ca n d vec d are distinct vectors such that vec axx vec c= vec bxx vec da n d vec axx vec b= vec cxx vec d , prove that ( vec a- vec d)dot (vec b- vec c)!=0,

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these

If vec ba n d vec c are two-noncollinear vectors such that vec a||( vec bxx vec c), then prove that ( vec axx vec b) . ( vec axx vec c) is equal to | vec a|^2( vec bdot vec c)dot

If vec a , vec ba n d vec c are unit coplanar vectors, then the scalar triple product [2 vec a- vec b2 vec b- vec c2 vec c- vec a] is a. 0 b. 1 c. -sqrt(3) d. sqrt(3)

If vec a , vec b ,a n d vec c are three vectors such that vec axx vec b= vec c , vec bxx vec c= vec a , vec cxx vec a= vec b , then prove that | vec a|=| vec b|=| vec c|dot