CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q222 Markscross_products
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Question 22 (Vector Algebra)
Find a vector of length 5 perpendicular to both (4,−1,3) and (−2,1,−2). OR A unit vector makes angles π/4, π/3 and acute θ with the coordinate axes. Find θ and its components.
Alternative 1 2 Marks
Find a vector of length 5 perpendicular to both (4,−1,3) and (−2,1,−2).
Answer: Either sign of (−5/3,10/3,10/3).
- The cross product is (−1,2,2).
- Its norm is 3, so divide by 3 to obtain a unit normal.
- Multiply by 5. Either orientation satisfies the stated vector conditions.
Alternative 2 2 Marks
A unit vector makes angles π/4, π/3 and acute θ with the coordinate axes. Find θ and its components.
Answer: θ=π/3, components (1/√2,1/2,1/2).
- The sum of squared direction cosines is one.
- Thus cos²θ=1−1/2−1/4=1/4. Take the positive cosine since θ is acute.
- Read off θ and the three cosines.