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Find the vector equation of a line which contains the two points $(1, 2, 3)$ and $(4, 5, 6) $

Hints only.

I know $x = a + t*d$ where $a, d$ are vectors and $t$ is a scalar.

We can have $a =\{<1, 2, 3>, <4, 5, 6> \}$ as two options of the vectors.

The difficulty is in finding $d$?

1 Answers 1

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Hint. You want both $u := (1,2,3)$ and $v: = (4,5,6)$ to lie on the line, that is for some values of $t_u, t_v$, we want $$ a + t_ud = u,\qquad a + t_vd = v $$ as we are free in choosing $t_u$ and $t_v$ (we just want them to lie on the line $a + \mathbf R d$ somewhere, let $t_u = 0$ and $t_1 = 1$, this gives $$ a = u, \qquad a + d = v $$ Can you continue from here?

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    Ah I see, so $d = v - u$? – 2017-01-08
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    But dont we need $t$ to be constant? – 2017-01-08
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    No, on the line $\ell: x = a + td$, $t$ runs through every real number and for each value gives a different point on $\ell$. And. Yes, $d = v-u$ is a possible choice. – 2017-01-08
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    Ah, so the variable is $t$? – 2017-01-08
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    Yes. That is part the definition of $x=a+td$ as the equation for the line: an equation that yields all points $x$ on the line, parameterized by the real number $t$.. Only $a$ and $d$ are constants. – 2017-01-08