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Let OP= α-1/α i+j+k, OQ= î+β-1/βj+k and OR=î+j+2k be three vectors, where
α,βR -{O} and O denotes the origin. If (OP×OQ). OR = 0 and the point (a, ß,2) lies on
the plane 3x+3y -z +l = 0, then the value of l is____.

Let p =2i + j +3k and q = i - j + k. If for some real numbers α, ß, and y, we have
15î+10j+6k=α(2p+q)+ß(p-2q)+y(pxq), then the value of y is____.

Let the position vectors of the points P,Q,R and S be a= 1+2j-5k, b=31+6j+3k, c=7i+16j+7k
and d= 2î + j+k, respectively. Then which of the following statements is true?
(A) The points P,Q,R and S are NOT coplanar
(B) b+2d/3 is the position vector of a point which divides PR internally in the ratio 5 : 4
(C) b+2d/3 is the position vector of a point which divides PR externally in the ratio 5 : 4
(D) The square of the magnitude of the vector bxd is 95

Let î, j and k be the unit vectors along the three positive coordinate axes.
ã=3î+j-k,
b=i+b2j+b3k,
č=c1+c2j+c3k,
be three vectors such that b2b3 greater than 0, a . b = 0

Let u, v and w be vectors in three-dimensional space, where u and v are unit vectors which are not
perpendicular to each other and u.w=1, v.w=1, w.w =4
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors u, v and w,
is v2 , then the value of 3u+5v is __

Let O be the origin and OA =2i+2j+k, OB= i-2j+2k and OC =1/2(OB-OA) for some a 0. If
|OBxOC|=9/2, then which of the following statements is (are) TRUE?

Let a and b be positive real numbers. Suppose PQ=ai+bj and PS=ai-bj are adjacent sides of a
parallelogram PQRS. Let u and v be the projection vectors of w = i + j along PQ and PS, respectively.
If u +v = w and if the area of the parallelogram PQRS is 8, then which of the following statements
is/are TRUE ?
(A) Projection of OCon OA is -3/2
(B) Area of the triangle OAB is 9/2
(C) Area of the triangle ABC is 9/2
(D) The acute angle between the diagonals of the parallelogram with adjacent sides OA and OC is π/3

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