3D Geometry
Planes and Angles between Planes
Grade 12
Question:
<p>Given the planes <i>x</i> + 2<i>y</i> − 3<i>z</i> + 5 = 0 and 2<i>x</i> + <i>y</i> + 3<i>z</i> + 1 = 0. If a point P is (2, −1, 2), then</p>
<p>(a) O and P both lie in acute angle between the planes</p>
<p>(b) O and P both lie in obtuse angle</p>
<p>(c) O lies in acute angle, P lies in obtuse angle</p>
<p>(d) O lies in obtuse angle, P lies in acute angle</p>
Step-by-Step Solution
Key Concept: A point lies in the acute angle between two planes if the product of the plane equations evaluated at that point is positive; obtuse angle if the product is negative.
Step 1: Check if origin O lies in acute or obtuse angle. Substitute O(0,0,0) in both planes: Plane 1: 0 + 0 − 0 + 5 = 5 > 0 Plane 2: 0 + 0 + 0 + 1 = 1 > 0 Both have same sign, so O lies in acute angle. Step 2: Check if P(2,−1,2) lies in acute or obtuse angle. Plane 1: 2 + 2(−1) − 3(2) + 5 = 2 − 2 − 6 + 5 = −1 < 0 Plane 2: 2(2) + (−1) + 3(2) + 1 = 4 − 1 + 6 + 1 = 10 > 0 Signs are opposite, so P lies in obtuse angle. ∴ Answer is (c).
Correct Answer: C