If c (vector b)- (vector a) = -3/4d (vector a) + 1/4d (vector b), can you conclude that d=4/3 and c=1/3 throug

Question

If c (vector b)- (vector a) = -3/4d (vector a) + 1/4d (vector b),
can you conclude that d=4/3 and c=1/3 through

direct comparison of the coefficients of vector a on the right hand side and the coefficients of vector a on the left hand side to find d

and then do the same for the coefficients of vector b?

Please explain clearly with some reasons. My teacher said this can not be done as you can't compare 4+2=10-6.

Answer 1

The simplest way is to put what belongs to vector b on one side, and what belongs to vector a on the other.

(c-d/4) vector b = (1 - 3d/4) vector a

Now, this equation is certainly true if we can find a condition for which both left- and right-hand sides equal 0.

Inspecting right-side gives 0 for d= 4/3, since it will give (1-1) vector a.
then, if you bring that value d in the (c-d/4) factor of the left-hand side, you get (c-1/3) vector b,
which in turn imposes c = 1/3 to nullify the left-hand side

Hence d= 4/3 and c=1/3.

Answer 2

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