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.

  • $\begingroup$ $r \times l=b$ 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

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...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.