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# how do I solve this?

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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 $$\left(\dfrac{c}{a}, \dfrac{e}{a}\right)$$?

May 4, 2020

#1
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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)

(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.

May 5, 2020
#2
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Yes that makes sense! Thank you!!

yeliah  May 5, 2020