CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Q355 Markscross_products
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Question 35 (Vector Algebra)
Nonzero vectors α,β,γ satisfy α×β=γ and β×γ=α. Show they are mutually perpendicular, norm β=1, and norm α=norm γ. Find the scalar projection of α on β. OR For A=(3,−1,2), B=(6,−3,6), C=(4,−3,1), D=(1,−1,−3), write their position vectors, prove ABCD is a parallelogram and find its area.
Alternative 1 5 Marks
Nonzero vectors α,β,γ satisfy α×β=γ and β×γ=α. Show they are mutually perpendicular, norm β=1, and norm α=norm γ. Find the scalar projection of α on β.
Answer: Mutually perpendicular; norm β=1; norm α=norm γ; projection zero.
- The first cross product makes γ perpendicular to α and β. The second makes α perpendicular to β and γ.
- Taking magnitudes gives norm γ=norm α·norm β and norm α=norm β·norm γ, since all mutual angles are right angles. Combine and cancel the nonzero norm α to get (norm β)²=1.
- Thus norm β=1 and norm α=norm γ. The projection is α·β/norm β=0.
Alternative 2 5 Marks
For A=(3,−1,2), B=(6,−3,6), C=(4,−3,1), D=(1,−1,−3), write their position vectors, prove ABCD is a parallelogram and find its area.
Answer: Area √165 square units.
- The position vectors are the listed coordinate triples. AB=(3,−2,4)=DC and AD=(−2,0,−5)=BC.
- Thus opposite directed sides agree. The cross product AB×AD=(10,7,−4), with squared norm 100+49+16=165.
- Therefore ABCD is a parallelogram of area √165 square units.