Question

The equation of the plane passing through the points (1, –2, 1), (2, –1, –3) and (0, 1, 5) is ______.

Options

• `bar"r"*(16hat"i" + 4hat"k")` = 10

• `bar"r"*(16hat"i" + 4hat"k")` = 20

• `bar"r"*(4hat"i" + 16hat"k")` = 20

• `bar"r"*(hat"i" + 16hat"k")` = 10

Answer 1

The equation of the plane passing through the points (1, –2, 1), (2, –1, –3) and (0, 1, 5) is  `bar"r"*(16hat"i" + 4hat"k")` = 20.

Explanation:

Let `bar"a", bar"b", bar"c"` be the position vectors of the points A, B, and C respectively.

∴ `bar"a" = hat"i" - 2hat"j" + hat"k", bar"b" = 2hat"i" - hat"j" - 3hat"k", bar"c" = hat"j" + 5hat"k"`

The vector equation of the plane passing through the points `"A"(bar"a"), "B"(bar"b")` and `"C"(bar"c")` is `bar"r"*(bar"AB" xx bar"AC") = bar"a"*(bar"AB" xx bar"AC")` 

`bar"AB" = bar"b" -  bar"a" =  hat"i" + hat"j" - hat"k"`

and `bar"AC" = bar"c" - bar"a" = -hat"i" + 3hat"j" + 4hat"k"`

∴ `bar"AB" xx bar"AC" = |(hat"i", hat"j", hat"k"),(1, 1, -4),(-1, 3, 4)|`

= `hat"i"(4 + 12) - hat"j"(4 - 4) + hat"k"(3 + 1)`

= `16hat"i" + 4hat"k"`

∴ `bar"r"*(16hat"i" + 4hat"k") = (hat"i" - 2hat"j" + hat"k")*(16hat"i" + 4hat"k")`

⇒ `bar"r"*(16hat"i" + 4hat"k") = 1(16) - 2(0) + 1(4)`

⇒ `bar"r"*(16hat"i" + 4hat"k")` = 20