Show that the line vector(r) = (2i - 2j + 3k) + λ (i - j + 4k).

Question

Show that the line \(\vec{r} = (2\hat{i}-2\hat{j}+3\hat{k}) + \lambda (\hat{i}-\hat{j}+4\hat{k}).\) is parallel to the plane \(\vec{r} . (\hat{i}+5\hat{j}+\hat{k}) = 5\) Also, find the distance between them.

Answer 1

As we know that the line \(\vec{r} = \vec{a} + \lambda \,\vec{b}\) is parallel to the plane \(\vec{r}.\vec{n}=q\) only when \(\vec{b}.\vec{n}=0\) and the distance between the line and the plane is given by

Hence, the given line is parallel to the given plane.

Now, distance between the given line and the given plane