Let P = (1,3) and Q = (2,6) be points in the plane. Find a vector-valued function R(t) = R0+tv such that R(t) describes the line through P and Q.

For the line l:x,y,z=5,3,-2+n1,2,-2, find a a point of the line. b a direction vector for the line.

Let L₁ be the line passing through the points Q₁(-4, 5, −5) and Q₂(−1, 6, −2) and let L2 be the line passing through the point P₁ (9, 11, 3) with direction vector d = [4, 2, 2] T. Determine whether L₁ and L₂ intersect. If so, find the point of intersection.

Q1. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, = (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them. Ans: 9x, - 12x, + 5х, - 5х, -0 Q2. If the vectors (0,1,a), (1, a,1) and (a,1,0) is linearly dependent, then find the value of a. Ans: 0,+/2 Q3. Find the eigen values and eigen vectors of the following matrices: 8 -6 2 (i) -6 7 [31 4] (ii) 0 2 6 0 o 5 -4 -4 3 Ans: (i) 0, 3, 15, k 2, ka (ii) 3, 2, 5, Q4. Verify Cayley-Hamilton theorem for the following matrix and hence compute A: [2 -1 1] A = -1 2 -1 I -1 2 [3 1 -1 Ans: 41 3 1 3 [2 11 Q5. Find the characteristic equation of the matrix A =0 1 0 and hence, compute A. Also find the matrix represented by A -5A" +7A“ - 3A +A* - SA' + 8A? - 2A +1. [8 5 5 [ 2 -1 -1 Ans: 2'-5a + 72 - 3 = 0, 0 3 0,A 3 55 8 3 -1 -1 10 5 Q6. Show that the matrix -2 -3 -4 has less than three linearly independent eigen vectors. Also 3 5 7 find them. Ans: A= 2,2,3. For i = 3, X, = [k,k,-2k] , for 2 = 2, X, = [5k,2k,-Sk] [i -1 2…