Question

The cosine of the angle included between the lines r = `(2hat"i" + hat"j" - 2hat"k") + lambda (hat"i" - 2hat"j" - 2hat"k")` and r = `(hat"i" + hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" - 6hat"k")` where λ, μ ∈ R is.

Options

• \[\frac{3}{21}\]

• \[\frac{17}{21}\]

• \[\frac{13}{21}\]

• \[\frac{11}{21}\]

Answer 1

\[\frac{11}{21}\]

Explanation:

Given lines,

r = `(2hat"i" + hat"j" - 2hat"k") + lambda (hat"i" - 2hat"j" - 2hat"k")`

r = `(hat"i" + hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" - 6hat"k")`

Here, `"b"_1 = hat"i" - 2hat"j" - 2hat"k"` and 

`"b"_2 = 3hat"i" + 2hat"j" - 6hat"k"`

`therefore cos theta = ("b"_1*"b"_2)/(|"b"_1| |"b"_2|)`

`= ((hat"i" - 2hat"j" - 2hat"k")*(3hat"i" + 2hat"j" - 6hat"k"))/((sqrt(1 + 4 + 4))(sqrt(9 + 4 + 36)))`

`cos theta = (3 - 4 + 12)/(3 xx 7)`

`= 11/21`