3. Find a vector that is orthogonal to both u = (2, 3, 1) and v = (0, 5, 7) usin

Question

3. Find a vector that is orthogonal to both u = (2, 3, 1) and v = (0, 5, 7) using cross
product.

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4. If u = (1, 2, 2), v = (3, 0, 4) and w = (6, 3, 2), then find 4 u v + 2w .

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5. Show that the set of vectors {(1, 2, 3), (2, 3, 1), (3, 2, 1)}, in R 3 , is linearly independent.

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6. Find the coordinate vector of w = (7, 5) relative to the basis {u 1 = (2, 4), u 2 = (3, 8)}
for R 2 .

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7.Find the dimension of the row space of A =

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8. If T (x 1 , x 2 , x 3 ) = (x 1 + x 2 , x 1 x 2 + 3x 3 ), then find [3]
(i) the domain of T .
(ii) the codomain of T .
(iii) the image of (3,4,2).

Explanation / Answer

given vectors u(2,_3,1) v(0,5,7)   

uxv = i j k

   2 -3 1

   0 5 7

= i(_21_5)_j(14_0)+k(10_0)

= 13i+7j_5k

check u.(uxv) = (2,-3,1) . (13,7,_5)

   = 0

v.(uxv) = (0,5,7).(13,7,_5)

   = 0

so,uxv is both orthogonal to u&v

given vectors u(1,-2,2) v(-3,0,4) w(6,-3,-2)

4u_v+2w

= 4(i_2j+2k) _j(_3i+0.j+4k) +2(6i_3j_2k)

=25i_14j

given vectors u1(2,_4) u2(3,8) w(7,5)

w=xu1+yu2

(7,5) = x(2,_4)+y(3,8)

=2x+3y=7

_4x+8y=5

x= 41/28 y=19/14

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