Question

Find the vector and cartesian equations of the line passing through the point (1, 2, –4) and perpendicular to the two lines

\(\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\) and \(\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}.\)

OR

Find the equation of a line passing through the point (1, 2, –4) and perpendicular to two lines

\(\vec{r} = (8\hat{i}-19\hat{j}+10\hat{k})+\lambda (3\hat{i}-16\hat{j}+7\hat{k})\) and \(\vec{r} = (15\hat{i}+29\hat{j}+5\hat{k})+\mu (3\hat{i}+8\hat{j}-5\hat{k}).\)

Answer 1

Let the cartesian equation of line passing through (1, 2, -4) be

Given lines are

Obviously parallel vectors \(\vec{b_1},​​\vec{b_2}\) and \(\vec{b_3}\) of (i), (ii) and (iii) respectively are given as

According to question

From equation (iv) and (v), we get

Putting the value of a, b, c in (i), we get the required cartesian equation of line as

Hence, vector equation is