Q. 1 - Calculus

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: For any two vectors u and v in ${\mathrm{ℝ}}^3$, $u\times v=v\times u$.

(b) True or False: If u and v are two vectors in ${\mathrm{ℝ}}^3$, then $u \times v = u · v$.

(c) True or False: If $u\times v = v\times u$, then u and v are parallel.

(d) True or False: If u, v, and w are vectors in R3, then (u × v) × w = u × (v × w).

(e) True or False: The triple scalar product can be used to find the volume of a parallelepiped.

(f) True or False: If u, v, and w are vectors in R3, then u · (v × w) = −v · (u × w).

(g) True or False: If u and v are nonparallel vectors in R3, then u · v u×v = cot θ, where θ is the angle between u and v.

(h) True or False: If u and v are unit vectors in R3, then u × v is also a unit vector.

Step-by-Step Solution

Answer

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Step 1: Read the Statement

We must determine if the statement is true or false:
True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.(a) True or False: For any two vectors u and v in ℝ3, u×v=v×u.(b) True or False: If u and v are two vectors in 

Step 2: Analyze the Statement

We examine the claim using relevant definitions, properties, or theorems.

Step 3: Conclusion

Ans: