Understanding Parallel Vectors: Solving for t with c = 3i + 4j and d = i -2j

[phospho]

Reading through a book and this question popped up, but the question didn't actually cover parallel vectors so I'm not sure what to make of it:

Given that c = 3i + 4j and d = i -2j find

t if d-tc is parallel to -2i + 3j

I done d - tc and got (1-3t)i + (-2-4t)j but I'm not quite sure how to equate this to the parallel vector -2i + 3j

in the solutions for some reason they have done the coefficient of i in vector -2i + 3j and multiplied it by the coeffecient of j in d-tc and done the same for j, so they have

-2(-2-4t) = 3(1-3t)

but I don't really know how they got that, if anyone could explain thanks.

 

[skiller]

If two vectors are parallel, what can you say about the coefficients of i and j ?

 

[phospho]

If two vectors are parallel, what can you say about the coefficients of i and j ?

I have no idea

edit: searched around and if two vectors are parallel then one of them is multiplied by a scalar, so

(1-3t)i + (-2-4t)j = k[-2i+3j]

so 1-3t = -2k , -2-4t = 3k

k = (1-3t)/-2

-2-4t = 3((1-3t)/-2)
t = -1/17

 

[skiller]

t = -1/17

Right!

Alternatively, you could say that they have the same "slope", so

(-2-4t) / (1-3t) = 3 / (-2)

ie the ratio of the i and j coefficients are the same.

(One has to be careful if one of the coefficients is zero, though.)

 

[phospho]

Right!

Alternatively, you could say that they have the same "slope", so

(-2-4t) / (1-3t) = 3 / (-2)

ie the ratio of the i and j coefficients are the same.

(One has to be careful if one of the coefficients is zero, though.)

thanks