Home
Class 12
MATHS
if overset(to)(b) " and " overset(to)(c...

if `overset(to)(b) " and " overset(to)(c )` are any two non- collinear unit vectors and `overset(to)(a)` is any vector then
`(overset(to)(a).overset(to)(b))overset(to)(b).(overset(to)(a).overset(to)(c )) overset(to)(c ) + .(overset(to)(a).(overset(to)(b)xxoverset(to)(c)))/(|overset(to)(b)xxoverset(to)(c)|^(2)).(overset(to)(b)xxoverset(to)(c))=.........`

Text Solution

Verified by Experts

The correct Answer is:
`overset(to)(a)`
Let `hat(i) ` be a unit vector in the direction of `vec(b) , hat(j)` in the direction of `vec(c )` . Note that `vec( c) = hat(j)`
and `(vec(b) xx vec(c )) = |vec(b)||vec(c )| " sin " alpha hat(k) = " sin " alpha hat(k)`
where `hat(k) ` is a vector perpendicular to `vec(b) " and " vec(c )`
`rArr |vec(b) xx vec(c )| " = sin " alpha rArr hat(k) = (vec(b) xx vec( c))/(|vec(b) xx vec(c )|)`
Let `vec( a) = a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k)`
Now `vec(a) ". " vec(b) = vec(a) ". " hat(i) = hat(i) "." (a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k)) = a_(1)`
and `vec(a) ". " vec(c ) = vec(a) " ." hat(j) = hat(j) ". " (a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k)) = a_(2)`
and `vec(a) ". " (vec(b) xx vec( c))/( |vec(b) xx vec( c)|) = vec(a) ". "vec(k) = a_(3) `
`:. (vec(a) ". " vec(b)) vec(b) + (vec(a) ". " vec(c )) vec( c) + (vec(a) (vec(b)xx vec(c )))/(|vec(b) xx vec(c )|^(2)) (vec(b) xx vec(c ))`
`=a_(1) vec(b) + a_(2) vec(c ) + a_(3) .((vec(b) xx vec(c )))/(|vec(b) xx vec(c )|) = a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) = hat(a)`
Promotional Banner

Topper's Solved these Questions

  • VECTOR ALGEBRA

    IIT JEE PREVIOUS YEAR|Exercise Vector Product of Two Vectors (Objective Questions I) (Analytical & Descriptive Questions)|4 Videos

  • VECTOR ALGEBRA

    IIT JEE PREVIOUS YEAR|Exercise Vector Triple Product (Objective Questions I)|9 Videos

  • VECTOR ALGEBRA

    IIT JEE PREVIOUS YEAR|Exercise Vector Product of Two Vectors (Objective Questions I) (Passage Based Problems)|2 Videos

  • TRIGONOMETRICAL RATIOS AND IDENTITIES

    IIT JEE PREVIOUS YEAR|Exercise HEIGHT AND DISTANCE|10 Videos

Similar Questions

Explore conceptually related problems

For any three vectors overset(to)(a), overset(to)(b) " and " overset(to)(C ) (overset(to)(a) - overset(to)(b)). {(overset(to)(b)-overset(to)(c))xx(overset(to)(c)-overset(to)(a))} = 2overset(to)(a).(overset(to)(b)xx overset(to)(c))

If overset(to)(A), overset(to)(B), overset(to)(C ) three non-coplanar vectors then (overset(to)(A) ,(overset(to)(B)xxoverset(to)(C)))/((overset(to)(C)xx overset(to)(A)). overset(to)(B))+ (overset(to)(B).(overset(to)(A) xx overset(to)(C)))/(overset(to)(C).(overset(to)(A)xx overset(to)(B)))=.........

For non- zero vectors overset(to)(a) , overset(to)(b), overset(to)(c )|,(overset(to)(a)xx overset(to)(b)), Overset(to)(c )| =|overset(to)(a)||overset(to)(b)||overset(to)(c )| holds if and only if

If overset(to)(a) , overset(to)(b) " and " overset(to)(c ) are three non- coplanar vectors then (overset(to)(a) + overset(to)(b) + overset(to)(c )) . [( overset(to)(a) + overset(to)(b)) xx (overset(to)(a) + overset(to)(c ))] equals

If vectors overset(to)(a) , overset(to)(b) , overset(to)( C) are coplanar then show that |{:(overset(to)(a),,overset(to)(b),,overset(to)(c )),(overset(to)(a)"."overset(to)(a),,overset(to)(a)"."overset(to)(b),,overset(to)(a)"."overset(to)(c )),(overset(to)(b)"."overset(to)(a),,overset(to)(b)"."overset(to)(b),,overset(to)(b)"." overset(to)(c )):}|

If overset(to)(a) , overset(to)(b) " and " overset(to)( c) are unit coplanar vectors then the scalar triple product [2 overset(to)(a) - overset(to)(b) 2 overset(to)(b) - overset(to)(c ) 2 overset(to)(c ) - overset(to)(a)] is

if overset(to)(a), overset(to)(b) " and " overset(to)(c ) are unit vectors satisfying |overset(to)(a)-overset(to)(b)|^(2)+|overset(to)(b)-overset(to)(c)|^(2)+|overset(to)(c)-overset(to)(a)|^(2)=9 |2overset(to)(a) +5overset(to)(b)+5overset(to)(c)| is equal to

The scalar overset(to)(A) .[(overset(to)(B) xx overset(to)( C)) xx (overset(to)(A) + overset(to)(B) + overset(to)( C))] equals

If the vectors overset(to)(b), overset(to)(c ) , overset(to)(d) are not coplanar then prove than the vectors (overset(to)(a) xx overset(to)(b)) xx (overset(to)(c ) xx overset(to)(d)) + (overset(to)(a) xx overset(to)(c )) xx (overset(to)(d) xx overset(to)(b)) +(overset(to)(a) xx overset(to)(d)) xx (overset(to)(b) xx overset(to)( c)) is parallel to overset(to)(a)