Math 225 Review for Exam 1
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Math 225
Review for Exam 1
1. Compute
⟨3, 0, −4⟩ · (⟨1, 1, 2⟩ × ⟨−1, 3, 0⟩).
2. If ⃗a = ⟨1, 0, 3⟩ and = ⟨2, 1, 7⟩, find a unit vector with positive first coordinate that is orthogonal to both ⃗a and .
3. Find an equation of the plane orthogonal to the line given by ⃗r(t) = ⟨6, 10, −1⟩+t⟨−9, −2, 8⟩ and containing the point (10, −1, 6).
4. What is the arc length from P = (0, 0, 3) to Q = (4π,0, 3) on ⃗r(t) = ⟨2t,3sin(2t), 3cos(2t)⟩?
5. Let ⃗r(t) = ⟨cos(2t), sin(2t), 1 + t⟩ . Write the accelaration function ⃗a(t) corresponding to ⃗r(t) as ⃗a(t) = aT (t)(t) + aN (t)Nˆ (t).
6. Compute the following
(a) t1〈 , , ln(t + 2)〉
(b) \1 t‘x2 ,esin(4/(x+1)),xln(x)〉dx
(c) (‘t − 1,t2 ,et〉בt−1 , cos(t), sin(t)〉)
(d) \〈t3 , − , cos2 (3t)〉 dt (e) \0 T/4‘4t,tan(t),et 〉dt
7. Let ⃗v = ⟨3, 2, −1⟩ and ⃗u = ⟨1, 1, 4⟩ . Compute proj⃗v (⃗u).
2023-02-10