Question

The equation of the plane passing through a point having position vector`-2hat"i" + 7hat"j" + 5hat"k"` and parallel to the vectors `4hat"i" - hat"j" + 3hat"k"` and `hat"i" + hat"j" + hat"k"` is ______.

Options

• `bar"r"*(4hat"i" + hat"j" + 5hat"k")` = 26

• `bar"r"*(4hat"i" - hat"j" + 5hat"k")` = 26

• `bar"r"*(-4hat"i" - hat"j" + 5hat"k")` = 26

• `bar"r"*(-4hat"i" - hat"j" + 5hat"k")` = 26

Answer 1

The equation of the plane passing through a point having position vector`-2hat"i" + 7hat"j" + 5hat"k"` and parallel to the vectors `4hat"i" - hat"j" + 3hat"k"` and `hat"i" + hat"j" + hat"k"` is `bar"r"*(-4hat"i" - hat"j" + 5hat"k")` = 26.

Explanation:

`bar"a" = -2hat"i" + 7hat"j" + 5hat"k"`

`bar"b"_1 = 4hat"i" - hatj" + 3hatk"`

`bar"b"_2 = hat"i" + hat"j" + hat"k"`

The given plane is perpendicular to the vector

`bar"n" - bar"b"_1 xx bar"b"_2`

= `|(hat"i", hat"j", hat"k"),(4, -1, 3),(1, 1, 1)|`

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

`bar"n"= -4hat"i" - hat"j" + 5hat"k"`

Vector equation of a plane is

`bar"r"*bar"n" = bar"a"*bar"n"`

∴ `bar"r"*(-4hat"i" - hat"j" + 5hat"k") = (-2hat"i" + 7hat"j" + 5hat"k")*(-4hat"i" -hat"j" + 5hat"k")`

= `(-2)(4) + (7)(-1) + (5)(5)`

∴ `bar"r"*(-4hat"i" - hat"j" + 5hat"k")` = 26