Question

A line passes through the points (6, −7, −1) and (2, −3, 1). Find the direction ratios and the direction cosines of the line. Show that the line does not pass through the origin.

Answer 1

A line passing through points A (6, −7, −1) and B (2, −3, 1) has direction ratios:

l = 2 − 6 = −4, m = −3 + 7 = 4, n = 1 + 1 = 2

Magnitude = `sqrt((-l)^2 + (m)^2 + (n)^2)`

= `sqrt((-4)^2+4^2+2^2)`

= `sqrt(16+16+4)`

= `sqrt36`

= 6

`cosalpha = (-4)/6 = -2/3, cosbeta = 4/6 = 2/3, cosgamma = 2/6 = 1/3`

The equation of a line passing through point A(6, −7, −1) with direction ratios (−4, 4, 2) is:

x = 6 − 4λ  ...(1)

y = −7 + 4λ  ...(2)

z = −1 + 2λ  ...(3)

To check if the line passes through the origin (0, 0, 0), we solve for λ:

Equation 1:

0 = 6 − 4λ

4λ = 6

λ = `6/4`

λ = `3/2`

Equation 2:

0 = −7 + 4λ

4λ = 7

λ = `7/4`

Equation 3:

0 = −1 + 2λ

2λ = 1

λ = `1/2`

For the line to pass through the origin, all three values of λ should be the same. But here, we obtained:

λ = `3/2`, λ = `7/4`, λ = `1/2`

Since these values are not equal, the line does not pass through the origin.