Question

The angle between the line `vec"r" = (5hat"i" - hat"j" - 4hat"k") + lambda(2hat"i" - hat"j" + hat"k")` and the plane `vec"r".(3hat"i" - 4hat"j" - hat"k") + 5` = 0 is `sin^-1 (5/(2sqrt(91)))`.

पर्याय

• True

• False

Answer 1

This statement is False.

Explanation:

Equation of line is `vec"r" = (5hat"i" - hat"j" - 4hat"k") + lambda(2hat"i" - hat"j" + hat"k")` and the equation of the plane is `vec"r" .(3hat"i" - 4hat"j" - hat"k") + 5` = 0

Here, `vec"b"_1 = 2hat"i" - hat"j" + hat"k"` and `vec"n"_2 = 3hat"i" - 4hat"j" - hat"k"`

∴ `sin theta = ("b"_1 vec"n"_2)/(|vec"b"_1||vec"n"_2|)`

⇒ `sin theta = ((2hat"i" - hat"j" + hat"k").(3hat"i" - 4hat"j" - hat"k"))/(sqrt(4 + 1 + 1)*sqrt(9 + 16 + 1)`

= `(6 + 4 - 1)/(sqrt(6)*sqrt(26))`

= `9/(sqrt(6)*sqrt(26))`

⇒ `sin theta = 9/(2sqrt(39)` which is false.