Please can someone help me solve this with step by step explanation?

9514 1404 393

  (b)  k = 3/4

  (c)  a = 1/13; b = 4/13

Step-by-step explanation:

(b) Parallel to (1, 1) means the two components of the vector are equal. (They have the same ratio as 1 : 1.) Solve by setting the components of the composite vector equal:

  p +kq = (1, 4) +k(3, -1) = (1 +3k, 4 -k)

We want ...

  1 +3k = 4 -k

  4k = 3 . . . . . . . add k-1 to both sides

  k = 3/4 . . . . . . divide by 4

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(c) Same as above: set the components of the composite vector to those of the desired sum.

  ap +bq = (1, 0)

  a(1, 4) +b(3, -1) = (1, 0)

  (a +3b, 4a -b) = (1, 0)

The second component tells us b=4a, so we can use that to find 'a' in the first component:

  a + 3(4a) = 1

  13a = 1

  a = 1/13

  b = 4a = 4/13

Check

  (1/13)(1, 4) + (4/13)(3, -1) = (1/13 +12/13, 4/13 -4/13) = (1, 0) . . . as required