These three points are on the same line: (1,2,3),(2,4,−6),(−3,−6,x). Find x. Explain your answer fully.

Question
Vectors
asked 2021-03-11
These three points are on the same line: (1,2,3),(2,4,−6),(−3,−6,x). Find x. Explain your answer fully.

Answers (1)

2021-03-12
You could do this using the vector equation of a line.
\(\displaystyle{r}{\left({t}\right)}={r}{0}+{t}{v}\)
We can use (1,2,3)(1,2,3) as the point r0 and (2,4,−6)−(1,2,3)=(1,2,−9) as the direction vector v. Then if (−3,−6,x)is on the line, it should be a solution r(t) to this equation. So let's see which xx this is possible for.
\(\displaystyle{\left(−{3},−{6},{x}\right)}={\left({1},{2},{3}\right)}+{t}{\left({1},{2},−{9}\right)}\)
\(\displaystyle{\left\lbrace-{3}={1}+{t},-{6}={2}+{2}{t},{x}={3}-{9}{t}\right\rbrace}\)
From the first equation, we see that t=−4t=−4. Hence, from the last equation, we get x=39
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