Consider the function f(x, y) 3-- 2 V Find Df(5, 5), where u = using each given vector v. (a) v i Duf(5, 5) (b) v -3i 4j Duf(5, 5) (c) v is the vector from (1, 2) to (-2, 6) Duf(5, 5) (d) v is the vector from (3, 2) to (4, 5). Duf(5, 5)

Find the B-coordinate vector for p(t) = -6+10t-7t² B={1-2t+t², 4-7t +5t², 5-8t+8t²} Enter your answer as (a,b,c) with no spaces between letters and commas.

Find two vectors v1 and v2 whose sum is (-5, -5, 5), where v1 is parallel to a =(0, –1, – 2) while v2 is perpendicular to a =(0, –1, –2). v1 = and v2 = Use "" signes to enteer a vector.

Express i = (3,7,-4) as a linear combination of i = (1,2,3), ỷ = (2,3,7), z = (3,5,6), also check whether the vectors i, ỹ, Z are linearly dependent or independent?

Find the vector equation of a line L going through the point P = (-5, -6) and parallel to the line generated by multiples of the vector v = --HA 1 L(t) - +t

Q1(A): [9 Marks] Two vectors A and B are located at P1(2,2,2) and P2(2,-2,-2). Write vector notation for each starting from the origin. What will be the distance between both vectors? Determine the dot product and the angle between them.

B1

Find the vectors u and v whose initial and terminal points are given. Initial Point Terminal Point u: (5, 1) (6, −2) v: (8, 8) (9, 5)

Find a non-zero vector that is orthogonal to (-7,6,5).   Separate your entries by commas, and don't forget the brackets.

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…