3D Geometry
Points and Planes
Grade 12
Question:
<p>The equation of a plane is \(2x - y - 3z = 5\) and \(A(1, 1, 1)\), \(B(2, 1, -3)\), \(C(1, -2, -2)\) and \(D(-3, 1, 2)\) are four points. Which of the following line segments are intersected by the plane?</p>
<p>(a) AD</p>
<p>(b) AB</p>
<p>(c) AC</p>
<p>(d) BC</p>
Step-by-Step Solution
Key Concept: A line segment intersects a plane if and only if its endpoints lie on opposite sides of the plane.
Step 1: For \(A(1, 1, 1)\): \(2(1) - 1 - 3(1) - 5 = 2 - 1 - 3 - 5 = -7 < 0\) Step 2: For \(B(2, 1, -3)\): \(2(2) - 1 - 3(-3) - 5 = 4 - 1 + 9 - 5 = 7 > 0\) Step 3: For \(C(1, -2, -2)\): \(2(1) - (-2) - 3(-2) - 5 = 2 + 2 + 6 - 5 = 5 > 0\) Step 4: For \(D(-3, 1, 2)\): \(2(-3) - 1 - 3(2) - 5 = -6 - 1 - 6 - 5 = -18 < 0\) Step 5: Points A and D are on one side (negative), while B and C are on the other side (positive). Therefore, line segments AB and AC intersect the plane. ∴ Answers are (b) and (c).
Correct Answer: b, c