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Answer:
(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
