3D Geometry
Equation of Plane
Grade 12
Question:
<p>The equation of plane containing line AC and at a maximum distance from B is</p><p>Given: A = 3<i>i</i> + 4<i>j</i>, C = 4<i>i</i> + 3<i>j</i>, B = 7<i>i</i> + 7<i>j</i></p>
<p>(a) <b>r</b> · (<i>i</i> + <i>j</i>) = 7</p>
<p>(b) <b>r</b> · (<i>i</i> − <i>j</i>) = 7</p>
<p>(c) <b>r</b> · (2<i>i</i> − <i>j</i>) = 7</p>
<p>(d) <b>r</b> · (3<i>i</i> + 4<i>j</i>) = 7</p>
Step-by-Step Solution
Key Concept: A plane containing a line and at maximum distance from an external point has a normal vector perpendicular to the line and pointing towards the external point.
Step 1: The plane containing line AC and at maximum distance from B must be perpendicular to the plane OABC. Step 2: Since OABC is a rhombus, the diagonal OB is perpendicular to AC. Step 3: Therefore, OB must be normal to the required plane. Step 4: The equation of plane is [ r − (4 i + 3 j )] · ( i + j ) = 0 Expanding: r · ( i + j ) = (4 i + 3 j ) · ( i + j ) = 4 + 3 = 7 ∴ Answer is (a).
Correct Answer: A