Vector Algebra
Vector Algebra
nta_pyq_2025_jan
Grade 12
Question:
Let c\to be the projection vector of \tob = \lambda i^ + 4k, ^ \to = i^ + 2 j^ + 2k \lambda > 0, on the vector a ^ \to + c\to| = 7, then the area . If |a of the parallelogram formed by the vectors \tob and c\to is ________ \to \to \to
Step-by-Step Solution
Key Concept: Apply the core result for dot product, cross product and projections and simplify using the given constraints.
c\to = ( \to \to ) \to b⋅a a \to \to | a| (16) | b| \lambda + 8 ^ ^ ^ = ( ) ( i + 2 j + 2k) 9 \to \to |a + c | = 7 \lambda + 8 2(\lambda + 8) 2(\lambda + 8) ∣ ^ ^ ^ \Rightarrow ( + 1) i + ( + 2) j + ( + 2) k∣ = 7 9 9 9 ∣ 2 2 2 \lambda + 8 2(\lambda + 8) 2(\lambda + 8) ( + 1) + ( + 2) + ( + 2) = 49 9 9 9 \Rightarrow \lambda = 4 \Rightarrow c = \to 4 3 ^ i + 8 3 ^ j + 8 ^ 3 k ∣ ^ ^ ^ ∣ i j k ∣ ∣ Area of parallelogram = ∣ 4 8 8 ∣ = 16 3 3 3 ∣ ∣ ∣ 4 0 4 ∣
Correct Answer: 16