Let's analyze the given statements to determine which ones are not true.

Question 10:

This question revolves around vector cross products and dot products in the context of vector functions $r(t)$, and unit vectors $T(t)$, $N(t)$, and $B(t)$, which often appear in the Frenet-Serret formulas in differential geometry.

• (A):

  • $[r′(t) × r′′(t)] · r(t) = [r(t) × r′(t)] · r′′(t)$
  • $[r′(t) × r′′(t)] · r(t) = [r′′(t) × r(t)] · r′(t)$

  This pair involves properties of vector triple products and scalar triple products. They are generally true based on standard vector identities.

• (B):

  • $[T(t) × N(t)] · B(t) = 1$
  • $[N(t) × T(t)] · B(t) = -1$

  Here, $T(t)$ is the tangent vector, $N(t)$ is the normal vector, and $B(t)$ is the binormal vector. According to the Frenet-Serret frame, $[T(t) × N(t)] · B(t)$ should indeed equal 1. The second expression $[N(t) × T(t)] · B(t)$ should equal 1 as well, because $[N(t) × T(t)] = B(t)$ and not $-B(t)$. Therefore, the second equation is false.

• (C):

  • $[B(t) × N(t)] · T(t) = 1$
  • $[N(t) × B(t)] · T(t) = -1$

  Similarly, by the Frenet-Serret identities, both expressions are expected to be true because of the relations between the vectors.

• (D):

  • $[B(t) × T(t)] · N(t) = 1$
  • $[T(t) × B(t)] · N(t) = -1$

  These are correct statements based on the properties of the cross and dot products in the Frenet-Serret frame.

Answer for Question 10: The false statement is (B).

Question 11:

This question involves vector triple products and their expansions, commonly found in vector calculus.

• (A):

  • $[r′(t) × r′′(t)] × r′(t) = |r′(t)|^2 r′′(t) + [r′(t) · r′′(t)]r′(t)$
  • $[r′(t) × r′′(t)] × r′′(t) = |r′′(t)|^2 r′(t) + [r′(t) · r′′(t)]r′′(t)$

  These equations are vector identities and they both are true based on standard vector calculus results.

• (B):

  • $[T(t) × N(t)] × B(t) = 0$
  • $[N(t) × T(t)] × B(t) = 0$

  This statement seems problematic. If $T(t)$, $N(t)$, and $B(t)$ form an orthonormal basis (as in the Frenet-Serret frame), the cross product of two should not be zero unless they are parallel or one of the vectors is zero. Since $T(t)$ and $N(t)$ are perpendicular, this expression is false.

• (C):

  • $[N(t) × B(t)] × T(t) = 0$
  • $[B(t) × N(t)] × T(t) = 0$

  Similarly to (B), these should not equal zero, making this pair false.

• (D):

  • $[B(t) × T(t)] × N(t) = 0$
  • $[T(t) × B(t)] × N(t) = 0$

  This pair is also false for the same reasons as (B) and (C).

Answer for Question 11: The false statements are (B), (C), and (D).

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Would you like a deeper explanation on any of the topics involved? Here are some questions to explore further:

1. What are the Frenet-Serret formulas?
2. How do vector triple products work in vector calculus?
3. What is the geometrical significance of the tangent, normal, and binormal vectors?
4. Can you prove the standard vector identities used here?
5. How does the cross product differ from the dot product in vector operations?
6. What is the significance of orthonormality in vectors?
7. What are the implications of vector identities in physics and engineering?
8. How do the cross and dot products relate to torque and angular momentum?

Tip: Always double-check vector identities by expanding the cross and dot products carefully, especially in applied mathematics contexts.