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Given the equation of a line in $\mathbb{R}^3$ as:

$$r \times l = b$$

Where $r$ is a general point of the line, $l$ is a unit vector along the direction of the line and $b$ is another vector. How can this form be converted to the vectorial form:

$$r = a + \lambda \cdot l$$

Where $r$ and $l$ convey the same meaning, $\lambda$ is a real parameter and $a$ is a specific point on the line. That is, given the cross product form, how can you find a specific point on the line.

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  • $\begingroup$ $r \times l=b$ describes a plane in $ \mathbb R^3$ and not a line ! $\endgroup$ – Fred Dec 9 '16 at 10:07
  • $\begingroup$ Doesn't $$ r \cdot n = c $$ describe a plane and not $$r×l=b$$? $\endgroup$ – user1611333 Dec 9 '16 at 10:10
  • $\begingroup$ Ooops ! you are right ! $\endgroup$ – Fred Dec 9 '16 at 10:12
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Take the cross product of LHS and RHS with $l$ and then apply the double cross product formula:

$$l \times (r \times l) = l \times b $$

$$(l \cdot l) r - (l \cdot r) l = l \times b $$

Up to you now...

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