A plane P1 has the equation 2 x-y+z=4 and the plane P2 has the equation x+n y+2 z=11 (where n ∈ N ). If the angle between P1 and P2 is (π/3) then the sum of all possible values of n, is

Question

A plane P1 has the equation 2 x-y+z=4 and the plane P2 has the equation x+n y+2 z=11 (where n ∈ N ). If the angle between P1 and P2 is (π/3) then the sum of all possible values of n, is (A) 1 (B) -16 (C) 17 (D) 16

Tardigrade Admin · Accepted Answer

h 1=2 hat i - hat j + hat k ; h 2= hat i + n hat j +2 hat k ; cos (π/3)=(2- n +2/√6 √5+ n 2)=(1/2) n 2+16 n -17=0 ⇒ n =-17 (reject as n ∈ N ) or n =1

Q. A plane $P_{1}$ has the equation $2 x - y + z = 4$ and the plane $P_{2}$ has the equation $x + n y + 2 z = 11$ (where $n \in N$ ). If the angle between $P_{1}$ and $P_{2}$ is $\frac{π}{3}$ then the sum of all possible values of $n$ , is

1

-16

17

16

$h_{1} = 2 \hat{i} - \hat{j} + \hat{k}; h_{2} = \hat{i} + n \hat{j} + 2 \hat{k}; cos \frac{π}{3} = \frac{2 - n + 2}{6 5 + n ^{2}} = \frac{1}{2}$
$n^{2} + 16 n - 17 = 0 \Rightarrow n = - 17$ (reject as $n \in N$ ) or $n = 1$

Q. A plane P1​ has the equation 2x−y+z=4 and the plane P2​ has the equation x+ny+2z=11 (where n∈N ). If the angle between P1​ and P2​ is 3π​ then the sum of all possible values of n, is

1

-16

17

16

h1​=2i^−j^​+k^;h2​=i^+nj^​+2k^;cos3π​=6​5+n2​2−n+2​=21​ n2+16n−17=0⇒n=−17 (reject as n∈N ) or n=1

Q. A plane $P_{1}$ has the equation $2 x - y + z = 4$ and the plane $P_{2}$ has the equation $x + n y + 2 z = 11$ (where $n \in N$ ). If the angle between $P_{1}$ and $P_{2}$ is $\frac{π}{3}$ then the sum of all possible values of $n$ , is

$h_{1} = 2 \hat{i} - \hat{j} + \hat{k}; h_{2} = \hat{i} + n \hat{j} + 2 \hat{k}; cos \frac{π}{3} = \frac{2 - n + 2}{6 5 + n ^{2}} = \frac{1}{2}$
$n^{2} + 16 n - 17 = 0 \Rightarrow n = - 17$ (reject as $n \in N$ ) or $n = 1$