Vector Algebra
Vector Algebra
nta_pyq_2025_jan
Grade 12
Question:
Let a \to = ^i + ^j + k, ^ \to \times b . If \to b = 2 i + 2 j + k and d = a ^ ^ ^ \to⋅\to c is a vector such that a \to \to \to| = 8 c = | c | , | c - 2a 2 \to \to \to \to \to \to 2 and the angle between d and c is \pi 4 , then |10 - 3 b ⋅ c | + | d \times c | is equal to
Step-by-Step Solution
Key Concept: Apply the core result for dot product, cross product and projections and simplify using the given constraints.
a ^ (6) \tob = 2 i^ + 2 j^ + k ^ \to d = a \times b \to \to ^ ^ = -i + j \to | c - 2a | \to 2 = 8 \to |c | 2 + 4|a| \to 2 \to \to - 4a ⋅ c = 8 \to |c | 2 + 12 - 4|c | = 8 \to \to |c | 2 \to - 4|c | + 4 = 0 \to =2 |c | 2 \to \to \times \tob d = a \to \to d \times c = (a \times b) \times c \to \to \to \to \to \to\to \to \to \to 2 \pi (|d | \times |c | sin \to ) = ((a ⋅ c ) b - ( b ⋅ c )a) 2 4 4 = 4| b| \to 2 \to \to \to 2 + ( b ⋅ c )2 (|a| ) - 2( b ⋅ c )(a ⋅ b) \to \to \to \to Let \tob ⋅ c\to = x 2 4 = 36 + 3x - 20x 2 3x - 20x + 32 = 0 8 x = ,4 3 \Rightarrow b ⋅ c = \to \to 8 ,4 3 \Rightarrow b ⋅ c = \to \to 8 3 Now, |10 - 3\tob ⋅ c\to| + |d\to \times c\to| 2 2 = |10 - 8| + (2) = 6
Correct Answer: 6