Question 1: Given vectors **A** = (3, 2) and **B** = (6, 4), determine if lines parallel when these vectors represent direction vectors of two lines.

Answer: Yes

Question 2: Construct a pair of parallel lines using vectors (2, 3) and (4, 6) as their direction vectors. Ensure the lines are distinct.

Answer: Students draw lines with these directions on the grid

Question 3: Calculate the equation of a line passing through point (1, 2) with a direction vector (3, 4).

Answer: y - 2 = (4/3)(x - 1)

Question 4: In a diagram, lines l and m are parallel, and line n cuts across them, forming alternate interior angles. Explain why the angles are equal using vector properties.

Answer: Because vectors are scalar multiples, they indicate parallel direction, hence the angles are equal by alternate interior angle theorem.

Question 5: Given a point (4, 5) and a vector (1, -2), write the parametric equations of the line.

Answer: x = 4 + t, y = 5 - 2t

Question 6: Prove that two lines with direction vectors (2, 1) and (4, 2) are parallel.

Answer: Vectors are scalar multiples, so the lines are parallel.

Question 7: A pair of lines are given with direction vectors (3, 4) and (6, 8). Are they parallel? Justify your answer.

Answer: Yes

Question 8: Identify and correct the error in the statement: 'Lines with direction vectors (1, 2) and (2, 4) are not parallel.'

Answer: They are parallel because vectors are scalar multiples.

Question 9: A line passes through (2, -1) with a direction vector (5, -10). Construct the line's equation and verify its parallelism with another line passing through (0, 0) with direction vector (1, -2).

Answer: Both lines are parallel since their vectors are scalar multiples.

Question 10: Explain how the scalar multiple property of vectors helps in establishing parallelism between lines.

Answer: Vectors are scalar multiples if and only if they are parallel, which implies the lines they direct are parallel.

Question 11: Challenge: Given two lines with direction vectors (7, 3) and (14, 6), show mathematically they are parallel, and then find a point on each line that are closest to each other.

Answer: Vectors are scalar multiples; the closest points can be found via projection or solving equations.