Question

Find the vector and cartesian equations of the line which is perpendicular to the lines with equations `(x + 2)/1 = (y - 3)/2 = (z + 1)/4` and `(x - 1)/2 = (y - 2)/3 = (z - 3)/4` and passes through the point (1, 1, 1). Also find the angle between the given lines.

Answer 1

Let us suppose the direction ratios of the required line L is a, b, c and it is perpendicular to the given lines

Then, a + 2b + 4c = 0   ...(i)

2a + 3b + 4c = 0   ...(ii)

Solving (i) and (ii) by cross multiplication

`a/(8 - 12) = b/(8 - 4) = c/(3 - 4)`

`a/(-4) = b/4 = c/(-1)`

∴ Direction ratios of line L are –4, 4, –1.

Then required vector and cartesian equations of the line L are respectively.

`vec(r) = hati + hatj + hatk + λ(-4hati + 4hatj - λ)`

And `(x - 1)/(-4) = (y - 1)/4 = (z - 1)/(-1)`

Now, Suppose θ is the angle between given lines,

So, `cos θ  = (|1 xx 2 + 2 xx 3 + 4 xx 4|)/(sqrt(1 + 4 + 16) sqrt(4 + 9 + 16))`

= `24/(sqrt(21) sqrt(29))`

∴ `θ = cos^-1 (24/sqrt(609))`