Question 1: Given vectors A = (3, 4) and B = (6, 8), determine whether A and B are parallel. Show your reasoning.

Answer: Yes, B is 2 times A, so vectors are parallel.

Question 2: Calculate a vector that is parallel to the vector (2, -3) with a scalar multiple of 4.

Answer: (8, -12)

Question 3: Plot the graph of y = 2x and construct a vector parallel to the line passing through the points (1, 2) and (3, 4).

Answer: Vector (2, 2)

Question 4: Construct a triangle with vertices at (1, 1), (4, 1), and (1, 4). Draw a vector parallel to the side from (1, 1) to (4, 1) starting at (2, 2).

Answer: Vector (3, 0) starting at (2, 2)

Question 5: Identify which of the following vectors are parallel: (5, 10) and (2, 4), (7, 14) and (3, 6), (1, 2) and (2, 5).

Answer: First two are parallel; third is not.

Question 6: A vector (x, y) is parallel to (4, -6). Find all possible vectors (x, y) where x and y are integers, with x between -10 and 10.

Answer: Any scalar multiple of (4, -6), e.g., (4, -6), (8, -12), (-4, 6), etc.

Question 7: If vector C = (k, 2k), for some scalar k, under what condition is C parallel to the vector (3, 6)?

Answer: When 3/1 = 6/2, so k = 3

Question 8: Determine the scalar multiple that makes vector (1, -2) parallel to (3, -6).

Answer: Scalar 3, since (3, -6) = 3*(1, -2)

Question 9: Explain why two vectors are parallel if one is a scalar multiple of the other.

Answer: Because they have the same or opposite direction, differing only in magnitude, which is represented by scalar multiplication.

Question 10: An engineer designs a vector (x, y) that is parallel to a known vector (5, -10). If the vector (x, y) has a magnitude of 10, find possible values for (x, y).

Answer: x= ±6.7, y= ∓13.4 (approximate, based on scalar multiple with magnitude 10)