Question 1: Express the translation of point P(3, 5) by the vector \\vec{v} = \\begin{bmatrix} 2 \\ -1 \\end{bmatrix} in coordinate form.

Answer: [5, 4]

Question 2: If a point Q has position vector \\vec{q} = \\begin{bmatrix} -1 \\ 4 \\end{bmatrix}, what is the position vector after translation by \\vec{u} = \\begin{bmatrix} 3 \\ -2 \\end{bmatrix}?

Answer: [2, 2]

Question 3: Write the vector notation for translating a point by 4 units right and 3 units up.

Answer: \\vec{v} = \\begin{bmatrix} 4 \\ 3 \\end{bmatrix}

Question 4: Construct and label a vector that moves a point 2 units left and 5 units down on the grid.

Answer: \\begin{bmatrix} -2 \\ -5 \\end{bmatrix}

Question 5: Given the point R(1, -3) and vector \\vec{w} = \\begin{bmatrix} -4 \\ 6 \\), find the coordinates of the translated point.

Answer: [-3, 3]

Question 6: A translation moves point A(7, 2) to point B(3, -1). Write the column vector representing this translation.

Answer: \\begin{bmatrix} -4 \\ -3 \\}

Question 7: If the translation vector is \\vec{t} = \\begin{bmatrix} 5 \\ 0 \\), what are the coordinates of the point after translating (2, 7)?

Answer: [7, 7]

Question 8: Explain what the notation \\vec{p} + \\vec{v} = \\begin{bmatrix} x + a \\ y + b \\} means in translation context.

Answer: It indicates translating a point with position vector \\vec{p} = \\begin{bmatrix} x \\ y \\} by the vector \\vec{v} = \\begin{bmatrix} a \\ b \\}, resulting in a new position vector.

Question 9: Identify and correct the mistake in the following notation: Moving point (4, 5) by vector \\begin{bmatrix} 2 \\ 3 \\} results in point (6, 8).

Answer: The correct translated point is (6, 8), so the notation is correct; the mistake is in assuming the translation vector results in (6, 8) from (4, 5).

Question 10: Challenge: Given points S(2, -1) and T(5, 4), find the translation vector that moves S to T. Write your answer in column vector notation.

Answer: \\begin{bmatrix} 3 \\ 5 \\}