Let p and q be unit vectors with an angle of 3 between them. If a vector r satisfies the relation r + 2( r p ) = q , then the value of 5| r |^2 is equal to:

Step-by-step solution

Given the vector relation: r + 2( r p ) = q Taking the dot product of both sides with p : r p + 2( r p ) p = q p Since ( r p ) p = 0 , we have: r p = q p The angle between the unit vectors p and q is 3 . Thus: r p = | q || p | ( 3 ) = (1)(1) ( 1 2 ) = 1 2 Now, squaring the original equation: | r + 2( r p )|^2 = | q |^2 | r |^2 + 4| r p |^2 + 4 r ( r p ) = 1 Since r ( r p ) = 0 , this simplifies to: | r |^2 + 4| r p |^2 = 1 Using Lagrange's identity, | r p |^2 = | r |^2| p |^2 - ( r p )^2 : | r |^2 + 4 (| r |^2(1)^2