Question 1: The resultant of two vectors is obtained by adding their corresponding components. If vectors \( \vec{A} = \begin{bmatrix} a_x \\ a_y \end{bmatrix} \) and \( \vec{B} = \begin{bmatrix} b_x \\ b_y \end{bmatrix} \), then the resultant vector \( \vec{R} \) is given by \( \vec{R} = \vec{A} + \vec{B} = \begin{bmatrix} a_x + b_x \\ a_y + b_y \end{bmatrix} \).

Answer: See full answer key

Question 1: Calculate the resultant of vectors \( \vec{A} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \) and \( \vec{B} = \begin{bmatrix} -1 \\ 5 \end{bmatrix} \).

Answer: \begin{bmatrix} 2 \\ 9 \end{bmatrix}

Question 2: Add the vectors \( \vec{C} = \begin{bmatrix} 5 \\ -2 \end{bmatrix} \) and \( \vec{D} = \begin{bmatrix} 2 \\ 7 \end{bmatrix} \).

Answer: \begin{bmatrix} 7 \\ 5 \end{bmatrix}

Question 3: What is the magnitude of the resultant vector when \( \vec{A} = \begin{bmatrix} 6 \\ 8 \end{bmatrix} \) and \( \vec{B} = \begin{bmatrix} -6 \\ -8 \end{bmatrix} \)?

Answer: 0

Question 4: Construct the vector resulting from adding \( \vec{E} = \begin{bmatrix} 4 \\ 1 \end{bmatrix} \) and \( \vec{F} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \) on the grid.

Answer: On the grid, draw from origin (0,0) to (4,1), then from (4,1) to (7,-1). The resultant vector points from (0,0) to (7,-1).

Question 1: Vectors \( \vec{G} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \) and \( \vec{H} = \begin{bmatrix} -4 \\ 1 \end{bmatrix} \) are added to find the resultant. If \( \vec{H} \) is scaled by a factor of 2 before addition, what is the resultant vector?

Answer: \begin{bmatrix} -0 \\ 5 \end{bmatrix}

Question 2: A boat moves with vectors \( \vec{V}_1 = \begin{bmatrix} 5 \\ 2 \end{bmatrix} \) east and \( \vec{V}_2 = \begin{bmatrix} 0 \\ 4 \end{bmatrix} \) north. Find the resultant displacement vector.

Answer: \begin{bmatrix} 5 \\ 6 \end{bmatrix}

Question 3: A student adds vectors \( \vec{J} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \( \vec{K} = \begin{bmatrix} 3 \\ -1 \end{bmatrix} \), but mistakenly subtracts \( \vec{K} \) instead of adding. Find the student's resultant and identify the error.

Answer: Incorrect result: \begin{bmatrix} -2 \\ 3 \end{bmatrix}. Error: subtracting instead of adding.

Question 4: In a real-world scenario, a hiker walks 3 km east and then 4 km north. Represent these movements as vectors and find the resultant displacement.

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

Question 1: Two vectors \( \vec{L} = \begin{bmatrix} 7 \\ -3 \end{bmatrix} \) and \( \vec{M} = \begin{bmatrix} -2 \\ 9 \end{bmatrix} \) are combined to produce a resultant vector. Find the magnitude and direction (angle with the positive x-axis) of the resultant.

Answer: Magnitude: \( \sqrt{5^2 + 6^2} = \sqrt{61} \), \( \approx 7.81 \). Direction: \( \tan^{-1}\left(\frac{6}{5}\right) \approx 50.2^\circ \).

Question 2: Construct a vector on the grid that is the resultant of \( \vec{N} = \begin{bmatrix} 4 \\ 7 \end{bmatrix} \) and \( \vec{O} = \begin{bmatrix} -3 \\ -2 \end{bmatrix} \), then verify your result by calculation.

Answer: Resultant vector: \( \begin{bmatrix} 1 \\ 5 \end{bmatrix} \).

Question 1: A student adds vectors \( \vec{P} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \) and \( \vec{Q} = \begin{bmatrix} 4 \\ 1 \end{bmatrix} \), but the answer obtained is \( \begin{bmatrix} 6 \\ 2 \end{bmatrix} \). What is the mistake? Correct the addition.

Answer: Mistake: student added the components correctly. The answer is correct. No mistake. Alternatively, if mistake: student incorrectly added components: e.g., added x-components as 6 (correct) but y-components as 2 (correct). So, no mistake here. If mistaken, the correction involves re-adding components.