00aut-m1sols

00aut-m1sols - MATH 51 MIDTERM 1 SOLUTIONS 1. Find all...

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MATH 51 MIDTERM 1 SOLUTIONS 1. Find all solutions of the following system: x 1 + x 2 + x 4 =7 x 1 + x 2 + x 3 + x 4 =1 0 x 1 + x 3 + x 4 =9 Solution. The augmented matrix for this system is 1101 1111 1011 ± ± ± ± ± ± 7 10 9 Its reduced row echelon form is 1001 0100 0010 ± ± ± ± ± ± 6 1 3 so the reduced form of the system is x 1 + x 4 =6 x 2 x 3 =3 Solving for the pivot variables x 1 ,x 2 3 we find x = x 1 x 2 x 3 x 4 = 6 - x 4 1 3 x 4 = 6 1 3 0 + x 4 - 1 0 0 1 2. Let L be the intersection of the two planes x +2 y +3 z =10 and 4 x +5 y +6 z =28 . Find a parametric equation for L . Solution. The augmented matrix for this system of equations is ² 123 456 ± ± ± ± 10 28 ³ Its reduced row echelon form is ² 10 - 1 01 2 ± ± ± ± 2 4 ³ 1
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so the system reduces to x - z =2 y +2 z =4 Solving for the pivot variables x and y gives x y z = 2+ z 4 - 2 z z = 2 4 0 + z 1 - 2 1 3. (a) Suppose u and v are vectors in R n such that u + v and u - v are orthogonal (i.e., perpendicular) to each other. Show that k u k = k v k . Solution. Two vectors are orthogonal if and only if their dot product is zero. So ( u + v ) · ( u - v )=0 Thele fthands ideexpandsto u · u - u · v + v · u - v · v = u · u - v · v = k u k 2 -k v k 2 Thus k u k 2 = k v k 2 ,so k u k = k v k . (b) Suppose u , v ,and w are unit vectors in R n . (Recall that a unit vector is a vector whose length is 1.) Suppose each vector is orthogonal (i.e., perpendicular) to each of the other two. Show that the two vectors u - 3 v w and u + v + w are orthogonal to each other. Solution. Since u , v and w are unit vectors, their lengths (and hence their lengths squared) are all equal to 1. So u · u = v · v = w · w = 1. Since each vector is perpendicular to the others, u · v = v · u =0 , u · w = w · u =0and v · w = w · v =0. So the dot product of the two vectors given is ( u - 3 v w ) · ( u + v + w )= u · u + u · v + u · w - 3 v · u - 3 v · v - 3 v · w w · u w · v w · w =1 - 3+2=0 so they are orthogonal. 4. Consider the points A =(1 , 1 , 1), B , 3 , 1) and C , 1 , 4) in R 3 .
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This note was uploaded on 01/12/2010 for the course MATH 51 at Stanford.

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00aut-m1sols - MATH 51 MIDTERM 1 SOLUTIONS 1. Find all...

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