A particle moves along the path $\vec{R}(t) = (2t^3+4t^2)\hat{i} + (3t^2-5t)\hat{j} + (7t^2+6t)\hat{k}$. Find the velocity component in the direction of $\hat{i} + \hat{j} + 3\hat{k}$ at $t = 1$:
Options:
(A) 122/√14
(B) 122
(C) 168/√14
(D) 168
Official Answer Key:
C
Step-by-Step Solution & Explanation
v⃗ = dR⃗/dt = (6t²+8t)î+(6t−5)ĵ+(14t+6)k̂. At t=2: v⃗=(40,7,34). Component along unit vector of (3,2,1) (|.|=√14): (40×3+7×2+34×1)/√14 = 168/√14.
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