Class 12 Three Dimensional Geometry · Very Long Answer5 Marksprojection_and_reflection
Class 12 Maths · Three Dimensional Geometry · Very Long Answer #52
For parallelogram A(4,7,8),B(2,3,4),C(−1,−2,1),D(1,2,5), find equations of all sides and the foot from A to CD.
Parallelogram and perpendicular foot on line CD
✅ Step-by-Step Solution
Final Answer: AB: r=A+t(1,2,2); BC: r=B+t(3,5,3); CD: r=C+t(1,2,2); DA: r=D+t(3,5,3). Foot (28/9,56/9,83/9). Cartesian sides: AB (x−4)=(y−7)/2=(z−8)/2; BC (x−2)/3=(y−3)/5=(z−4)/3; CD (x+1)=(y+2)/2=(z−1)/2; DA (x−1)/3=(y−2)/5=(z−5)/3.
A+C=B+D=(3,5,9), confirming the parallelogram. Compute each adjacent displacement.
Side equations are r=A+t(1,2,2), r=B+t(3,5,3), r=C+t(1,2,2), r=D+t(3,5,3); t∈ℝ separately. Cartesian forms follow from these nonzero directions. Cartesian sides: AB (x−4)=(y−7)/2=(z−8)/2; BC (x−2)/3=(y−3)/5=(z−4)/3; CD (x+1)=(y+2)/2=(z−1)/2; DA (x−1)/3=(y−2)/5=(z−5)/3.
For H=C+s(1,2,2), s=((A−C)·(1,2,2))/9=(5+18+14)/9=37/9.
Thus H=(28/9,56/9,83/9). The dot product (A−H)·(1,2,2)=0 verifies the foot.
📋 CBSE Stepwise Marking Scheme
[1 mark] A+C=B+D=(3,5,9), confirming the parallelogram. Compute each adjacent displacement.
[1 mark] Side equations are r=A+t(1,2,2), r=B+t(3,5,3), r=C+t(1,2,2), r=D+t(3,5,3); t∈ℝ separately. Cartesian forms follow from these nonzero directions. Cartesian sides: AB (x−4)=(y−7)/2=(z−8)/2; BC (x−2)/3=(y−3)/5=(z−4)/3; CD (x+1)=(y+2)/2=(z−1)/2; DA (x−1)/3=(y−2)/5=(z−5)/3.
[1 mark] For H=C+s(1,2,2), s=((A−C)·(1,2,2))/9=(5+18+14)/9=37/9.
[1 mark] Thus H=(28/9,56/9,83/9). The dot product (A−H)·(1,2,2)=0 verifies the foot.
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