INTRODUCTION TO LINEAR ALGEBRA TEST #4 SELECT AND WORK ON ANY 4 PROBLEMS cross p

Question

INTRODUCTION TO LINEAR ALGEBRA TEST #4 SELECT AND WORK ON ANY 4 PROBLEMS cross product find the area of the triangle with vertices A(-1,2,-3); B(-4,-2,-1) and #2. Let i, j and k be the three unit coordinate vectors (pairwise orthogonal vectors, each of norm. magnitude 1). Compute the following norm and cross products. (a) 121-4j 4k|1 (b) ixj, ixk, jxk (c) (31-2j + 3k) (-2i +3j-3k #3 Find all 2 2 matrices for which x-G) is an eigenvector with associated eigenvalue-2. #4. For which values of the constant k does the matrix A-(k 2 3 #5. Find all eigenvalues of the matrix A-1 1 0 1 #6. If A is 2x2 matrix with det(A)- -5 and tr(A)-4, what are the eigenvalues of A? hx2) is an eigenvector with associated eigenvalue X 2 ) have-2 as an eigenvalue?

Explanation / Answer

1) The vertices are given as : A ( -1,2,-3) B (-4,-2,-1) and C (1,-6,-9)
Thus, the vector AB will be = OB-OA
= -4i - 2j - k - ( -i +2j - 3k) = (-4+1) i + (-2 -2 ) j + (-1+3) k = -3i -4j +2k
Thus, vector AB = -3i - 4j + 2k

Similarly vector AC = OC-OA =
= i -6j - 9k - ( -i + 2j - 3k ) = 2i - 8j - 6k

Thus, area of a triangle ABC can be given as a vector cross product as = 1/2 | AB X AC |
= 1/2 | i j k |
-3 -4 2
2 -8 -6  
= 1/2 | [ i ( 24 - (-16) ) - j ( 18 - 4) + k ( 24 - (-8) ) ] |
= 1/2 |[ 40 i - 14 j + 32k ] |
= 1/2 ( sqrt ( 402 + (-14)2 + 322 ) )
= 1/2 sqrt ( 1600 + 196 + 1024 )
= 1/2 sqrt ( 2820 )
= 2 * 1/2 * sqrt (705)
= sqrt (705)
Thus, area of the traingle is sqrt (705) sq units

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