Question

If a = `3hat"i" + hat"j" - hat"k"`, b = `2hat"i" - hat"j" + 7hat"k"` and c = `7hat"i" - hat"j" + 23 hat"k"` are three vectors, then which of the following statement is true.

Options

• a and b are collinear

• a, b, c are mutually perpendicular

• a, b and c are coplanar

• a, b and c are non-coplanar

Answer 1

a, b and c are non-coplanar

Explanation:

Given, vectors

a = `3hat"i" + hat"j" - hat"k"`, b = `2hat"i" - hat"j" + 7hat"k"` and

c = `7hat"i" - hat"j" + 23 hat"k"`

a ≠ λb

∴ a and b are not collinear

∴ Option (a) is false.

a . b = 6 - 1 - 7 ≠ 0

∴ a and b are not perpendicular

∴ a, b, c are not mutually perpendicular

∴ Option (b) is also incorrect.

`"a" * ("b" xx "c") = |(3,1,-1),(2,-1,7),(7,-1,23)|`

= 3(- 23 + 7) - 1 (46 - 49) - 1 (- 2 + 7)

= - 48 + 3 - 5 = - 50 ≠ 0

∴ a , b and c are non-coplanar

∴ Option (d) IV is true.