Find the co-ordinates of the point where the line vector (r) = (- i - 2j - 3k) + λ (3i + 4j + 3k)

Question

Find the co-ordinates of the point where the line \(\vec{r}=(-\hat{i}-2\hat{j}-3\hat{k}) + \lambda (3\hat{i}+4\hat{j}+3\hat{k})\) meets the plane which is perpendicular to the vector \(\vec{n}=\hat{i}+\hat{j}+3\hat{k}\) and at a distance of \(\frac{4}{\sqrt{11}}\) from origin.

Answer 1

We know that the equation of plane is

\(=\vec{r}.\hat{n}\, d;\) where is normal unit vector and d is perpendicular distance from origin.

\(\therefore\) Equation of plane

Equation of given line

Its cartesian form is

Let Q \((\alpha, \beta, \gamma)\) be the point of intersection of (i) & (ii)

\(\because\) Q lies on (ii)

Also, Q lies on (i)

\(\therefore\) Required point of intersection \(\equiv\) (2, 2, 0)