The number of distinct values of `lamda`, for which the vectors `-lamda^(2)hati+hatj+hatk, hati-lamda^(2)hatj+hatk` and `hati+hatj-lamda^(2)hatk` are coplanar, is

Options

A. zero
B. one
C. two
D. three

Answer

C

Explanation

We know that three vectors are coplanar if their scalar triple product is zero. Thus,
`|{:(-lamda^(2),,1,,1),(1,,-lamda^(2),,1),(1,,1,,-lamda^(2)):}| =0 R_1 to R_1 + R_2 + R_3`
or `" "|{:(2-lamda^(2),,2-lamda^(2),,2-lamda^(2)),(1,,-lamda^(2),,1),(1,,1,,-lamda^(2)):}|=0`
or `" "(2-lamda)|{:(1,,1,,1),(1,,-lamda^(2),,1),(1,,1,,-lamda^(2)):}|=0`
or `" "(2-lamda^(2)) |{:(1,,1,,1),(0,,-(1+lamda^(2)),,0),(0,,0,,-(1+lamda^(2)):}|=0`
`" "(R_2 to R_2 - R_1, R_3to R_3 - R_1)`
or `(2-lamda^(2))(1+lamda^(2))^(2) =0 rArr lamda = pm sqrt2`
Hence, the real solutions.