Vector Algebra

For real numbers , , , and , consider the matrix M = bmatrix & 1 2 & - 1 2 \ 1 3 & & 1 3 \ & & bmatrix . Suppose that MMT = I , where MT is the transpose of the matrix M, and I is the 3 3 identity matrix. Let u = i + 1 3 j + k , v = 1 2 i + j + k and w = - 1 2 i + 1 3 j + k . Match each entry in List-I to the correct entry in List-II and choose the correct option.

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

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

For any two points M and N in the XY-plane, let MN denote the vector from M to N, and 0 denote the zero vector. Let P, Q and R be three distinct points in the XY-plane. Let S be a point inside the triangle PQR such that SP + 5 SQ + 6 SR = 0. Let E and F be the mid-points of the sides PR and QR, respectively. Then the value of length of the line segment EFlength of the line segment ES is .

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)

Consider the vectors x= i+2 j+3 k, y=2 i+3 j+ k, z=3 i+ j+2 k. For two distinct positive real numbers and , define X= x+ y- z, Y= y+ z- x, Z= z+ x- y. If the vectors X, Y, and Z lie in a plane, then the value of + -3 is .

Let OP= -1 i+ j+ k, OQ= i+ -1 j+ k and OR= i+ j+ 12 k be three vectors, where , R-\0\ and O denotes the origin. If ( OP OQ ) OR=0 and the point ( , ,2) lies on the plane 3x+3y-z+l=0, then the value of l is .

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 .