Question

If the points having the position vectors `2hat"i" + hat"j" - hat"k", -hat"j", 4hat"i" + 4hat"k"` and `lambdahat"i" + hat"k"` lie on the same plane, then λ is equal to ______.

Options

• – 1

• – 2

• 1

• 2

Answer 1

If the points having the position vectors `2hat"i" + hat"j" - hat"k", -hat"j", 4hat"i" + 4hat"k"` and `lambdahat"i" + hat"k"` lie on the same plane, then λ is equal to 2.

Explanation:

Let `bar"a" = 2hat"i" + hat"j" - hat"k", bar"b" = -hat"j",`

`bar"c" = 4hat"i" + 4hat"k"` and `bar"d" = lambdahat"i" + hat"k"`

`bar"AB" = -2hat"i" - 2hat"j" + hat"k",`

`bar"AC" = 2hat"i" - hat"j" + 5hat"k"`

`bar"AD" = (lambda - 2)hat"i" - hat"j" + 2hat"k"`

Since the given points are coplanar,

`[(bar"AB", bar"AC", bar"AD")]` = 0

⇒ `|(-2, -2, 1),(2, -1, 5),(lambda - 2, -1, 2)|` = 0

⇒ `-2(-2 + 5) + 2(4 - 5lambda + 10) + 1(-2 + lambda - 2)` = 0

⇒ λ = 2