Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsVector Algebra

Let p and q be two vectors such that | p |=2 and | q |= 5 . If the projection of p on q is 2 5 , and the position vectors of the vertices of a quadrilateral PQRS are 2 p + q , 5 p +2 q , 4 p +6 q and p +4 q respectively, then the area (in sq. units) of the quadrilateral PQRS is equal to :

Options

  1. A96
  2. B32
  3. C24
  4. D48

Correct answer

D. 48

Step-by-step solution

The projection of p on q is given by p q | q | . p q 5 = 2 5 p q = 2 Using Lagrange's identity, | p q |^2 = | p |^2| q |^2 - ( p q )^2 | p q |^2 = (2)^2( 5 )^2 - (2)^2 = 20 - 4 = 16 | p q | = 4 The diagonal vectors of the quadrilateral PQRS are : PR = (4 p +6 q ) - (2 p + q ) = 2 p +5 q QS = ( p +4 q ) - (5 p +2 q ) = -4 p +2 q The area of the quadrilateral PQRS is 1 2 | PR QS | . PR QS = (2 p +5 q ) (-4 p +2 q ) = 4( p q ) - 20( q p ) = 4( p q ) + 20( p q ) = 24( p q ) Area = 1 2 |24( p q )| = 12| p q | Area = 12

Practice Vector Algebra on Quantrex Academy →

More from Vector Algebra

Let a , b be two vectors, and let P, Q and R be the points with position vectors a , b and a + b , respectively, with respect to the origin O . If | a + b | = 21 , | a - b | = 3 , 2026Let a = 4 i - j + 3 k , b = 10 i + 2 j - k and a vector c be such that 2( a b ) + 3( b c ) = 0 . If a c = 15 , then c ( i + j -3 k ) is equal to: 2026Let a = 2 i + 3 j + 3 k and b = 6 i + 3 j + 3 k . Then the square of the area of the triangle with adjacent sides determined by the vectors (2 a + 3 b ) and ( a - b ) is : 2026If a = i + j + k , b = j - k and c be three vectors such that a c = b and a c = 3 , then c ( a - 2 b ) is equal to _______. 2026Let O be the origin, OP = a and OQ = b . If R is the point on OP such that OP = 5 OR , and M is the point such that OQ = 5 RM , then PM is equal to : 2026Let a = 7 i + j - k and b = j + 2 k . If r is a vector such that r a + a b = 0 and r a = 0 , then |3 r |^2 is equal to: 2026Let u and v be unit vectors inclined at an acute angle such that | u v |= 3 2 . If A = u + v +( u v ) , then is equal to: 2026Let a_k = ( _k) i + j and b_k = i - ( _k) j , where _k = 2^ k-1 2^n + 1 , for some n N , n > 5 . Then the value of _ k=1 ^ n | a_k |^2 _ k=1 ^ n | b_k |^2 is _____. 2026 Full Vector Algebra list All JEE Main PYQs