Question 1: Given vectors A = (3, 6) and B = (6, 12), determine if they are parallel.

Answer: Yes, B = 2A, so they are parallel.

Question 2: Calculate a scalar multiple k such that the vector (4, 8) is a scalar multiple of (2, 4).

Answer: k = 2

Question 3: Are the vectors (1, 2) and (2, 5) parallel? Justify your answer.

Answer: No, because 5/2 ≠ 2/1, so not scalar multiples.

Question 4: Construct a vector that is parallel to (3, -3) with a magnitude of 6 units.

Answer: Vector (6, -6) or (-6, 6).

Question 1: Vectors P = (2, 4) and Q = (k, 8) are parallel. Find the value of k and explain how you know they are parallel.

Answer: k = 2, because Q = 2P, so they are scalar multiples.

Question 2: A boat moves from point A (3, 5) to point B (9, 11). Determine a vector representing this movement. If a second boat moves from C (x, y) to D, and AD is parallel to AB, find possible coordinates for D if the movement is scaled by a factor of 3.

Answer: AB = (6, 6); scaled by 3 gives (18, 18); D could be at (x + 18, y + 18).

Question 3: Determine whether the vectors (5, 10) and (3, 6) are parallel. Justify your answer.

Answer: Yes, because 10/5 = 2 and 6/3 = 2, so scalar multiples.

Question 4: Construct a vector parallel to (-4, 7) with a magnitude of 10 units.

Answer: Approximate vector: (-4/√65)*10, (7/√65)*10.

Question 1: A cyclist travels from point X (2, 3) to point Y (8, 7). A second cyclist starts at point Z (x, y) and travels along a route parallel to XY, ending 15 units from Z in the same direction. Find possible coordinates for Z.

Answer: Vector XY = (6, 4); scaled to 15 units gives magnitude; Z at (x, y) such that vector ZD is parallel and length 15.

Question 1: A vector A = (x, y). A student claims that any scalar multiple of A is only parallel to A if |k| > 1. Is this true? Explain and correct the misconception.

Answer: False. Scalar multiples are always parallel regardless of the scalar's magnitude.

Question 2: Given vectors U = (1, 2) and V = (4, 8), and a third vector W = (5, 10). Identify which are parallel, and explain why some are not.

Answer: U, V, and W are all parallel because each is a scalar multiple of (1, 2).

Question 1: A student states: 'Vectors (2, 4) and (3, 6) are parallel because their x-components are proportional.' Is this correct? Justify your answer.

Answer: No, because their y-components are not proportional (4/2=2, 6/3=2), but the statement is generally correct; the key is both components proportionally, so yes, they are parallel.

Question 2: A vector (a, b) is parallel to (c, d). If a ≠ 0, what is the condition relating a, b, c, d to ensure parallelism?

Answer: b/a = d/c, assuming c ≠ 0.