Q1. Note: Attempt all questions. The marks for each question are indicated against it.
- a)) Three vectors ā, b and ĉ satisfy the condition a+b+c=0. If |d| = 4, |b| = 2 and |c| = 3 determine the value of a.b+b.c + c.a. (150 words)
- b)) A curve is described by the following parametric equations x = 2 cost, y = 2sint, z = cos(3t) Determine the unit tangent vector to the curve at the point t = π/2. (150 words)
Key Points:
- If `a+b+c=0`, then `(a+b+c)² = 0` is used to find scalar products.
- The identity `(a+b+c)² = |a|² + |b|² + |c|² + 2(a.b + b.c + c.a)` is crucial.
- The position vector `r(t)` describes the path of a particle in parametric form.
- The derivative `dr/dt` (velocity vector) gives the tangent vector to the curve.
Answer: This response comprehensively addresses two distinct problems related to vector algebra and vector calculus, fundamental concepts in mechanics. Part (a) focuses on manipulating vector dot products when their sum is zero, deriving a relationship between the magnitudes and the sum of dot products. Part (b) involves vector differentiation to find the tangent vector of a parametrically defined curve and then normalizing it to obtain the unit tangent vector at a specific point. Both solutions demonst...