Question 1: Calculate the magnitude of the vector v = (3, 4).

Answer: 5

Question 2: A vector has components (−6, 8). Find its magnitude.

Answer: 10

Question 3: Calculate |v| for v = (−5, −12).

Answer: 13

Question 4: If a vector has components (x, y) and |v| = 7, what is the relationship between x and y? (Express as an equation or inequality.)

Answer: x² + y² = 49

Question 5: A student calculates |v| = √(x² + y²) with x=3 and y=4, but writes the answer as 7. Identify and correct the mistake.

Answer: Incorrect calculation of √(3² + 4²) = √(9 + 16) = √25 = 5, not 7.

Question 6: Construct a vector with components (x, y) such that |v| = 10, and both x and y are integers.

Answer: Examples include (6, 8), (8, 6), (−6, 8), etc.

Question 7: A vector has components (9, −12). Find its magnitude and explain any common mistakes made in the calculation.

Answer: |v| = √(81 + 144) = √225 = 15. Common mistake: mixing signs or miscomputing squares.

Question 8: Plot the graph of y=2x on the grid and determine the magnitude of the vector from the origin to the point (3, 6).

Answer: |v| = √(3² + 6²) = √(9 + 36) = √45 ≈ 6.7

Question 9: Construct a triangle with vertices at the origin (0,0), (x, 0), and (0, y) such that the hypotenuse has length 13 and x,y are positive integers. Find possible values of (x, y).

Answer: Examples include (5, 12) and (12, 5).

Question 10: Explain why the formula |v| = √(x² + y²) is valid for calculating the magnitude of a vector in the coordinate plane.

Answer: Because the magnitude is the length of the hypotenuse of a right triangle with legs x and y, by Pythagoras’ theorem.

Question 11: A vector has components (−8, −15). Find its magnitude and discuss the common errors students make in such calculations.

Answer: |v| = √(64 + 225) = √289 = 17. Common error: neglecting the squares or adding directly.