Question

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

\(\frac{x-8}{8}=\frac{y+9}{-16}=\frac{z-10}{7}\) and \(\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}\)

Answer 1

The Cartesian equation of a line passing through a point (x₁, y₁, z₁) and having directional ratios proportional to a, b, c is given by,

\(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\)

The required line passes through the point (1, 2, –4), now we need to find the direction ratios of the line which are a, b, c . this equation of the required line is,

\(\frac{x-1}{a}=\frac{y-2}{b}=\frac{z+4}{c}\)

It is given that a line having Cartesian equation \(\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}\) is perpendicular to the required line,

So the dot product equation will be equal to zero.

a × 3 + b × 8 + c ×(–5) = 0

3a + 8b – 5c = 0 ……..(ii).

By solving equation (i) and (ii), we get, by using cross multiplication method,

Put these values in the required equation of line,

Therefore, this is the required equation of line.