Let a =a₁ i +a₂ j +a₃ k where a ₁, a ₂, a ₃ and | a | are rational numbers. If a makes an angle 45^ with b and b = 2 i +3 2 j +4 k then a lies in

Step-by-step solution

Given a =a₁ i+a₂ j +a₃ k and b = 2 i+3 2 i+4 k Where, a ₁, a ₂, a ₃ and | a | are rational numbers. Now acc to question aligned & ( A 2 ) ( c 2 )= 2 a b+a^2+b^2-c^2 2 b c-b^2-c^2+a^2 & = (2 b) a+ (a^2-c^2 )+b^2 (2 b) c-b^2-c^2+a^2 & = (a+c) a+a^2-c^2+ (a+c)^2 4 (a+c) c- (a+c)^2 4 -c^2+a^2 aligned 2 a ₁+6 a ₂+4 3 a ₃=| a | .6 Since a₁ a₂, a₃,|a| are rational numbers thus both sides should be rational a₃=0 hence a lies in XY place. Now ( A 2 ) ( c 2 )= 2 ab + a ^2+ b ^2- c ^2 2 bc - b ^2- c ^2+ a ^2 aligned & = (2 b)