Question 1: Two vectors are parallel if one is a scalar multiple of the other. That is, vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if \( \mathbf{a} = k \mathbf{b} \) or \( \mathbf{b} = k \mathbf{a} \) for some scalar \( k \).

Answer: Vectors are parallel if one is a scalar multiple of the other.

Question 2: Calculate whether the vectors \( \mathbf{u} = (3, 6) \) and \( \mathbf{v} = (1, 2) \) are parallel.

Answer: Yes, \( \mathbf{u} = 3 \times \mathbf{v} \), so they are parallel.

Question 3: Given vectors \( \mathbf{p} = (4, -2) \) and \( \mathbf{q} = (-2, 1) \), determine if they are parallel.

Answer: Yes, \( \mathbf{q} = -0.5 \times \mathbf{p} \), so they are parallel.

Question 4: Plot the vectors \( \mathbf{a} = (2, 4) \) and \( \mathbf{b} = (1, 2) \) on the grid. Are they parallel?

Answer: Yes, \( \mathbf{a} = 2 \times \mathbf{b} \), so they are parallel.

Question 1: Determine if \( \mathbf{m} = (5, 10) \) and \( \mathbf{n} = (1, 2) \) are parallel.

Answer: Yes, \( \mathbf{m} = 5 \times \mathbf{n} \).

Question 2: Find a scalar \( k \) such that \( (7, -14) \) is parallel to \( (1, -2) \).

Answer: The scalar \( k = 7 \).

Question 3: Prove that the vectors \( (x, 3x) \) and \( (2x, 6x) \), for any real \( x \), are parallel.

Answer: Since \( (2x, 6x) = 2 \times (x, 3x) \), they are scalar multiples, hence parallel.

Question 4: Construct on the grid a vector \( \mathbf{c} = (4, 8) \) and a vector \( \mathbf{d} = (2, 4) \). Are they parallel?

Answer: Yes, \( \mathbf{c} = 2 \times \mathbf{d} \).

Question 1: Vectors \( \mathbf{p} = (x, 2x) \) and \( \mathbf{q} = (3, 6) \) are parallel. Find the value of \( x \) and justify your answer.

Answer: Since \( (3, 6) = 3 \times (1, 2) \), \( \mathbf{p} \) is parallel when \( x = 1 \).

Question 2: Explain why vectors \( (5, 10) \) and \( (2.5, 5) \) are parallel, using the scalar multiple concept.

Answer: Because \( (2.5, 5) = 0.5 \times (5, 10) \), they are scalar multiples, hence parallel.

Question 3: A boat moves from point A to point B, with vector \( \mathbf{v} = (6, 12) \). If the current's velocity vector is \( \mathbf{w} = (3, 6) \), is the current affecting the boat's direction? Justify your answer.

Answer: Yes, \( \mathbf{w} \) is a scalar multiple of \( \mathbf{v} \), meaning the current flows in the same or opposite direction, affecting the boat's movement.

Question 1: A truck moves along a straight path represented by \( \mathbf{t} = (50, 0) \) km. A cyclist travels along the same line with vector \( \mathbf{c} = (25, 0) \) km. Are their routes parallel? Justify your answer.

Answer: Yes, \( \mathbf{c} = 0.5 \times \mathbf{t} \), so their routes are along the same line, hence parallel.

Question 1: Given vectors \( \mathbf{r} = (x, 2x + 1) \) and \( \mathbf{s} = (4, 9) \), determine for which \( x \) they are parallel. Show all steps.

Answer: Set \( (4, 9) = k \times (x, 2x+1) \). From first component: \( 4 = kx \). From second: \( 9 = k(2x+1) \). Solving leads to \( x = 2 \).

Question 2: Construct a vector \( \mathbf{u} = (x, 3x) \) and a vector \( \mathbf{v} = (9, 27) \). Find all possible \( x \) for which they are parallel.

Answer: They are parallel when \( \mathbf{v} = 3 \times \mathbf{u} \), so \( x \) can be any real number.

Question 1: Student's statement: 'Vectors \( (2, 4) \) and \( (4, 8) \) are not parallel because they are different vectors.' Is this correct? Explain your answer.

Answer: Incorrect. They are parallel since \( (4,8) = 2 \times (2,4) \).

Question 2: A student claims that \( (1, 2) \) and \( (2, 4) \) are not parallel because they are not equal. Is this correct? Why or why not?

Answer: Incorrect. Vectors can be parallel even if they are not equal; they are scalar multiples.