2. Determine all vectors v that are orthogonal to (2,-b,0. a) (8,2,t), where r i

Question

2. Determine all vectors v that are orthogonal to (2,-b,0. a) (8,2,t), where r is any real number b) (6r, 41,$), where s and r are any real numbers c) d) c) v = (81. 4r,s), where s and are any real numbers v = (6t.21.,), where r is any real number 3. Find (u,v) for the inner product (u.v)=3u,v, +»,v, defined in R, where u-(-6,2) and vs(0,4) a) s b) 36 c) 12 d) 32 e) 16 4. Use the functions f (x)-x', and g (x)-x1 +1 in C [-1,1] to find (f, g) for the inner product (ra)=j/(x)g(x)dr (.s) 15 a) 56 b) = 255 (.g) 15 112 d) 255 (f.g) 16 85 c) Use the inner product (4.B) = 2a,i bi-a 2h, + aiba + 2a22h22 to find A, B) , where 5. 3-3|and B =1-5-1 a) 4, B)-39 b) 4, B)-39 (4, B)-33 d) (4,B)--17 (4.B) -17 e)

Explanation / Answer

Answer(2):

Given that u and v are orthogonal then uv=0
Ans(a):
given v=(8t,2t,t)
uv=0
(2,-6,0)*(8t,2t,t)=0
16t-12t+0=0
4t=0
t=0
plug value of t into v=(8t,2t,t) we get v=(0,0,0)
hence required answer is v=(0,0,0)

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Ans(b):
given v=(6t,4t,s)
uv=0
(2,-6,0)*(6t,4t,s)=0
12t-24t+0=0
-12t=0
t=0
plug value of t into v=(6t,4t,s) we get v=(0,0,s)
there is no equation left to find value of s
so s is arbitrary means for any value of s, v will be orthogonal to u.
Hence required answer is v=(0,0,s)

-----------------------------
Ans(c):
given v=(8t,4t,s)
uv=0
(2,-6,0)*(8t,4t,s)=0
16t-24t+0=0
-8t=0
t=0
there is no equation left to find value of s
so s is arbitrary means for any value of s, v will be orthogonal to u.
plug value of t into v=(8t,4t,s) we get v=(0,0,s)
Hence required answer is v=(0,0,s)

-----------------------------
Ans(d):
given v=(6t,2t,t)
uv=0
(2,-6,0)*(6t,2t,t)=0
12t-12t+0=0
0=0
which is true so t is arbitrary means for any value of t, v will be orthogonal to u.
Hence required answer is v=(6t,2t,t)

-----------------------------
Ans(e):
given v=(6t,2t,s)
uv=0
(2,-6,0)*(6t,2t,s)=0
12t-12t+0=0
0=0
which is true so t is arbitrary means for any value of t, v will be orthogonal to u.
there is no equation left to find value of s
so s is arbitrary means for any value of s, v will be orthogonal to u.
Hence required answer is v=(6t,2t,s)

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