Vector Algebra
Vector Algebra
nta_pyq_2025_jan
Grade 12
Question:
If the components of \to a = \alpha i + \beta j + \gamma k along and perpendicular to b = 3 i + j - k respectively, are ^ ^ ^ ^ ^ ^ 16 11 ^ ^ ^ (3 i + j - k) and 1 11 ^ ^ ^ (-4 i - 5 j - 17k) , then \alpha + \beta + \gamma is equal to : 2 2 2
26
18
23
16 \to and \tob be two unit vectors such that the angle between them is
Step-by-Step Solution
Key Concept: Apply the core result for dot product, cross product and projections and simplify using the given constraints.
let (1) \to a11 = component of a along b \to \to \to a1 = \to perpendicular to \tob component of a \to 16 ^ ^ ^ a 11 = (3 i + j - k) 11 \to 1 ^ ^ ^ a 1 = (-4 i - 5 j - 17k) 11 \to \to \to ∵ a = a 11 + a 1 \to \therefore a = 16 1 ^ ^ ^ ^ ^ ^ = (3 i + j - k) + (-4 i - 5 j - 17k) 11 11 \to 33 ^ ^ 11 j - k 11 ^ ^ ^ \alpha = 4 i + j - 3k 2 2 2 \alpha + \beta + \gamma = 16 + 1 + 9 = 26
Correct Answer: 1