Test on A vector and its coordinates

Question 1: For points P(1, 2, 3) and Q(4, 6, 5), the coordinates of vector **PQ** are found by subtracting the coordinates of P from the coordinates of Q, resulting in (3, 4, 2).

A. Yes

B. No

Explanation: The study materials state that if a vector connects two arbitrary points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), its coordinates are found by subtracting the coordinates of the initial point from the coordinates of the terminal point: **PQ** = (x₂ - x₁, y₂ - y₁, z₂ - z₁). Given P(1, 2, 3) and Q(4, 6, 5), the calculation is (4 - 1, 6 - 2, 5 - 3), which results in (3, 4, 2). This aligns with the example provided in the materials.

Question 2: According to the study materials, which statement accurately describes the characteristics of free vectors and position vectors?

A. Free vectors must always have their initial point at the origin of a coordinate system.

B. Position vectors define the position of their terminal point relative to the origin.

C. Any two oriented line segments with different lengths but the same direction can represent the same free vector.

D. A vector is considered a position vector if its magnitude is always greater than one.

Explanation: The study materials state that 'for convenience, vectors are often placed with their initial point at the origin (0, 0, 0) of a coordinate system. These are called position vectors, as they define the position of their terminal point relative to the origin.' This directly supports option 1. Option 0 is incorrect because free vectors are explicitly stated as 'not tied to a specific starting point.' Option 2 is incorrect because free vectors require the same length and direction. Option 3 introduces a criterion (magnitude greater than one) not mentioned in the context of defining a position vector.

Question 3: A point (3, 4, 2) and a vector (3, 4, 2) fundamentally represent the same concept in a coordinate system.

A. Yes

B. No

Explanation: A point specifies a location, whereas a vector specifies a displacement, direction, and magnitude. Although a position vector can use the coordinates of its terminal point, the underlying concepts of a point as a location and a vector as a movement are distinct.

Question 4: If vector **c** has coordinates (-2, 5, -8), its opposite vector -**c** has coordinates (2, 5, 8).

A. Yes

B. No

Explanation: For any vector **v** with coordinates (vₓ, vᵧ, v_z), its opposite vector -**v** has coordinates (-vₓ, -vᵧ, -v_z). Therefore, if vector **c** is (-2, 5, -8), its opposite vector -**c** must have coordinates (-(-2), -5, -(-8)), which simplifies to (2, -5, 8). The statement incorrectly implies that the y-component remains the same and the z-component changes sign twice.

Question 5: According to the study materials, which of the following is a common error when calculating the magnitude of a vector?

A. Forgetting to take the square root at the end of the calculation.

B. Incorrectly squaring negative components (e.g., treating (-3)² as -9 instead of 9).

C. Summing the components before squaring them.

D. Confusing a vector's components with its magnitude directly.

Explanation: The study materials explicitly list 'Incorrect Magnitude Calculation' as a common error, stating: 'A common error is forgetting to take the square root at the end of the magnitude calculation or incorrectly handling negative components (e.g., (-3)² should be 9, not -9). Always square each component before summing them.' Therefore, options 0 and 1 directly reflect these stated common errors. Options 2 and 3 are not mentioned as common magnitude calculation errors in the provided text.