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

Sum

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

#### Solution

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"/|(-1, -2),(1, -2)| = "c"/|(-2,1),(-3,-4)|

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

∴ "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.

Concept: Vector Product of Vectors (Cross)
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