Question

The vector equation of the plane r = `(2hat"i" + hat"k") + lambda(hat"i") + mu(hat"i" + 2hat"j" - 3hat"k")` in scalar product form is `"r"*(3hat"i" + 2hat"k") = alpha`, then α = ______.

पर्याय

• 2

• 3

• 1

• 0

Answer 1

The vector equation of the plane r = `(2hat"i" + hat"k") + lambda(hat"i") + mu(hat"i" + 2hat"j" - 3hat"k")` in scalar product form is `"r"*(3hat"i" + 2hat"k") = alpha`, then α = 2.

Explanation:

Given, equation of plane

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

Here, plane (i) passing through a (let) = `2hat"i" + hat"k"`

and parallel to vector b (let) = `hat"i"` and

c = `hat"i" + 2hat"j" - 3hat"k"`

We know that equation of plane passing through a point a and parallel to non-parallel vectors band c is

r · (b × c) = a · (b × c) = [a b c]

Now, [a b c] = `|(2,0,1),(1,0,0),(1,2,-3)|`

= 2(0) - 0 + 1(2 - 0) = 2

and `"b" xx "c" = |(hat"i", hat"j", hat"k"),(1,0,0),(1,2,-3)| = 3hat"i" + 2hat"k"`

∴ `"r" * (3hat"i" + 2hat"k") = 2`

Therefore, α = 2