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> + 2 = 0 and <i>x</i> − 2<i>y</i> + 3<i>z</i> + 7 = 0, if the point P is (1, 2, 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: When the plane equations evaluated at a point have the same sign, the point lies in the acute angle between the two planes.
Step 1: Check if origin O lies in acute or obtuse angle. Plane 1 at O: 0 + 0 − 0 + 2 = 2 > 0 Plane 2 at O: 0 − 0 + 0 + 7 = 7 > 0 Both positive, so O lies in acute angle. Step 2: Check if P(1,2,2) lies in acute or obtuse angle. Plane 1 at P: 1 + 2(2) − 3(2) + 2 = 1 + 4 − 6 + 2 = 1 > 0 Plane 2 at P: 1 − 2(2) + 3(2) + 7 = 1 − 4 + 6 + 7 = 10 > 0 Both positive, so P lies in acute angle. ∴ Answer is (a).
Correct Answer: A