Let a 1 x + b 1 y + c 1 z + d 1 = 0 and a 2 x + b 2 y + c 2 z + d 2 = 0 be two planes, where d 1 , d 2 > 0. Then origin lies in acute angle if a 1 a 2 + b 1 b 2 + c 1 c 2 < 0 and origin lies in obtuse angle if a 1 a 2 + b 1 b 2 + c 1 c 2 > 0.
Further point (x 1 , y 1 , z 1 ) and origin both lie either in acute angle or in obtuse angle ,
if (a 1 x 1 + b 1 y 1 + c 1 z 1 + d 1 ) (a 2 x 1 + b 2 y 1 + c 2 z 1 + d 2 ) > 0, one of (x 1 , y 1 , z 1 ) and origin lie in acute angle and the other in obtuse angle, if (a 1 x 1 + b 1 y 1 + c 1 z 1 + d 1 ) (a 2 x 1 + b 2 y 1 + c 2 z 1 + d 2 ) < 0
(i) Given the planes 2x + 3y – 4z + 7 = 0 and x – 2y + 3z – 5 = 0, if a point P is (1, – 2, 3) and O is origin, then
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i)
Equation of the second plane is –x + 2y –3z + 5 = 0 ⇒ 2 (–1) + 3 . 2 + (– 4) (–3) > 0
∴ O lies in obtuse angle (2 × 1 + 3(–2) – 4 × 3 + 7) (–1 + 2 (–2) – 3 × 3 + 5)
= (2 – 6 – 12 + 7) (–1 – 4 – 9 + 5) > 0 ∴ P lies in obtuse angle.
(ii)
1 × 2 + 2 × 1 – 3 × 3 < 0 ∴ O lies in acute angle.
Also (2 + 2 (–1) – 3(2) + 5) (2 × 2 – 1 + 3 × 2 + 1) = (–1) (10) < 0 ∴ P lies in obtuse angle.
(iii)
1 – 4 – 9 < 0 ∴ O lies in acute angle.
Further (1 + 4 – 6 + 2) (1 – 4 + 6 + 7) > 0 ∴ The point P lies in acute angle.
──────────────────────────────────────────────────────────────────────────────────────────
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems