For what value of t are these two vectors parallel?
a) r = 4i + tj and s = 14i - 12j
What I Know: vectors are parallel if one is a scalar multiple of the other. I need to find a way that r and s relate by finding what number you can multiply r by to equal s.
What I Don't Know: How to approach the problem. How can 14i be related to 4i?
so you want:
(14, - 12) = k(4, t)
that is, 14 = 4k and -12 = kt
4k = 14
k = 14/4 = 7/2
then kt = -12
(7/2)t = -12
t = -12(2/7) = -24/7
Never mind! I got it
LCM(4,14) = 28
So,
7r = 28i + 7tj
2s = 28i - 24j
so, if 7t = -24, 2s = 7r, so one is a multiple of the other.
Or, more directly,
14/4 r = 14i + 14/4 t j
so, if 14/4 t = -12, s is a multiple of r.
To determine if two vectors are parallel, we need to find a scalar multiple that relates one vector to the other. In this case, we want to find a value of t that makes r a scalar multiple of s.
We have vector r = 4i + tj and vector s = 14i - 12j.
To see if r and s are scalar multiples of each other, we can compare the components of the two vectors.
Comparing the i-components, we have 4 = 14t. Dividing both sides by 14, we get t = 4/14 = 2/7.
Comparing the j-components, we have t = -12t. Dividing both sides by -12, we get t = -12/-12 = 1.
Therefore, for t = 2/7 or t = 1, the vector r = 4i + tj is parallel to the vector s = 14i - 12j.