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, and a and ( a- b) are perpendicular to each other, then the area of the triangle OPR is

1. 3
2. 32
3. 3 32
4. 32

Let w = i + j - 2 k, and u and v be two vectors, such that u v = w and v w = u. Let , , , and t be real numbers such that u = i + j + k, -t + + = 0, - t + = 0, and + - t = 0. Match each entry in List-I to the correct entry in List-II and choose the correct option.

1. (P) → (2), (Q) → (1), (R) → (4), (S) → (5)
2. (P) → (2), (Q) → (4), (R) → (3), (S) → (5)
3. (P) → (2), (Q) → (1), (R) → (4), (S) → (3)
4. (P) → (5), (Q) → (4), (R) → (1), (S) → (3)

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