Question 1: Write the vector equation of a line parallel to the line passing through points A(2, 3) and B(5, 7).

Answer: r = (2,3) + t(3,4)

Question 2: Calculate the vector difference between points C(4, 5) and D(1, 2).

Answer: (3, 3)

Question 3: Determine if the vectors (2, 4) and (4, 8) are parallel. Show your working.

Answer: Yes, (4/2) = (8/4) = 2, so parallel.

Question 4: Using vector algebra, prove that lines with parallel direction vectors are parallel.

Answer: If two lines have proportional direction vectors, then their direction ratios are scalar multiples, hence lines are parallel.

Question 5: Construct a line passing through point P(3, 2) parallel to the line passing through Q(1, 1) and R(4, 5).

Answer: r = (3, 2) + t(3, 4)

Question 6: Given vectors u = (1, 2) and v = (2, 4), determine whether the lines with these direction vectors are parallel.

Answer: Yes

Question 7: A line passing through point (2, 3) is parallel to a line with direction vector (6, 8). Find the equation of the line in vector form.

Answer: r = (2, 3) + t(3, 4)

Question 8: Explain why parallel lines have equal corresponding angles when cut by a transversal.

Answer: Because the alternate interior angles are equal, and parallel lines maintain consistent angles with a transversal.

Question 9: A transversal cuts two lines, forming alternate interior angles of 70°. Are the lines parallel? Show your reasoning.

Answer: Yes, because alternate interior angles are equal, so lines are parallel.

Question 10: Identify the common mistake students might make when determining if two vectors are parallel and explain how to correct it.

Answer: Mistake: Checking only one component. Correction: Verify proportionality of all components.

Question 11: Challenge: Prove that if the dot product of two direction vectors is zero, the lines are perpendicular.

Answer: If u·v = 0, then u and v are orthogonal, so the lines are perpendicular.