Question

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are - 2, 1, - 1 and - 3, - 4, 1

Answer 1

Let a, b, c be the direction ratios of the vector which is perpendicular to the two lines whose direction ratios are -2, 1, -1 and -3, -4, 1

∴ - 2a + b - c = 0  and - 3a - 4b + c = 0

∴ `"a"/|(1,-1),(-4, 1)| = (-"b")/|(-2, -1),(-3, 1)| = "c"/|(-2,1),(-3,-4)|`

∴ `"a"/(1 - 4) = (-"b")/(-2 - 3) = "c"/(8 + 3)`

∴ `"a"/-3 = (-"b")/(-5) = "c"/11`

∴ `"a"/-3 = "b"/5 = "c"/11`

∴ the required direction ratios are  - 3, 5, 11

Alternative Method:

Let `bar"a"` and `bar"b"` be the vectors along the lines whose direction ratios are -2, 1, -1 and -3, -4, 1 respectively.

Then `bar"a" = - 2hat"i" + hat"j" - hat"k"` and `bar"b" = - 3hat"i" - 4hat"j" + hat"k"`

The vector perpendicular to both `bar"a"` and `bar"b"` is given by

`bar"a" xx bar"b" = |(hat"i",hat"j",hat"k"),(-2, 1, -1),(-3, -4, 1)|`

`= (1 - 4)hat"i" - (- 2 - 3)hat"j" + (8 + 3)hat"k"`

`= - 3hat"i" + 5hat"j" + 11hat"k"`

Hence, the required direction ratios are - 3, 5, 11.