Question

Solve the following:

Find the acute angle between the line `bar"r" = lambda(hat"i" - hat"j" + hat"k")` and the plane `bar"r".(2hat"i" - hat"j" + hat"k")` = 23.

Answer 1

The acute angle θ between the line `bar"r" = bar"a" + lambdabar"b"` and and the plane `bar"r".bar"n" = d` is given by

sin θ = `|(bar"b".bar"n")/(|bar"b"||bar"n"|)|`       ...(1)

Here, `bar"b" = hat"2i" + hat"3j" - hat"6k", bar"n" = 2hat"i" - hat"j" + hat"k"`

∴ `bar"b".bar"n" = (hat"2i" + hat"3j" - hat"6k").(2hat"i" - hat"j" + hat"k")`

= (2)(2) + (3)(– 1) + (– 6)(1)
= 4 – 3 – 6
= – 5
Also, `|bar"b"| = sqrt(2^2 + 3^2 + (– 6)^2) = sqrt(49)` = 7

`|bar"n"| = sqrt(2^2 + (– 1)^2 + 1^2) = sqrt(6)`

∴ from (1), we have

sin θ = `|(-5)/(7sqrt(6))| = 5/(7sqrt(6))`

∴ θ = `sin^-1 ((5)/(7sqrt(6)))`.