+76 Consider the line through points \(A = (1, 1, 2)\)and \(B = (2, 3, 4)\).
This line can be parametrized as \((at +b, ct +d, et +f)\)for some choices of constants \(a,b, c, d, e\) and \(f\). What is the ordered pair \(\left(\dfrac{c}{a}, \dfrac{e}{a}\right)\)?
+499 in general, a line given two vectors can be parametrized as :
a + t*d
where:
a = starting vector
t = any real number
d = the direction vector
In this case, we have (1,1,2) and (2,3,4)
the direction vector can be found through subtracting the two. We get:
(2,3,4) - (1,1,2) = (1,2,2)
with our starting point being (1,1,2)
We then get the line equation as:
(1,1,2) + t(1,2,2)
this then turns into:
(1,1,2) + (t,2t,2t)
Adding the two, we get:
(t+1,2t+1,2t+2)
a = 1
b = 1
c = 2
d = 1
e = 2
f = 2
c/a = 2/1 = 2
e/a = 2/1 = 2
the ordered pair (c/a, e/a) is then (2,2). Please correct me if I'm wrong, precalc has never been one of my strong suits.
