Vector Algebra
Vector Algebra
nta_pyq_2025_jan
Grade 12
Question:
Let a b = 3 i + j - k and c ^ ^ ^ \to and \tob . If the vector C\to \to be three vectors such that c\to is coplanar with a is perpendicular to \tob and a \to ⋅ c\to = 5, then |c\to| is equal to
\sqrt 11 6
1 3\sqrt2
16
18
Step-by-Step Solution
Key Concept: Apply the core result for dot product, cross product and projections and simplify using the given constraints.
\to \to c = \lambda( b \times ( a \times b )) (1) \to \to \to \to \to \to = \lambda(( b ⋅ b ) a - ( a ⋅ b ) b ) \to \to = \lambda(11 a - 2 b ) = \lambda(11i + 22j + 33k - 6i - 2j + 2k) = \lambda(5i + 20j + 35k) = 5\lambda(5i + 4j + 7k) \to \to = Given c ⋅ a = 5 1 = 5\lambda(1 + 8 + 21) = 5 = \lambda = 30 \to 1 \Rightarrow c = (i + 4j + 7k) 6 \to \sqrt1 + 16 + 49 11 c = = \sqrt 6 6
Correct Answer: 1