This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨5, 0⟩ and v = ⟨-2, 0⟩, we compute u + v = ⟨5 + (-2), 0 + 0⟩ = ⟨3, 0⟩. Choice A is correct because it shows the component-wise calculation with specific values. Choice C subtracts the vectors instead of adding them, calculating ⟨5 - (-2), 0 - 0⟩ but mistakenly as ⟨-3, 0⟩, which is not the sum. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions. Remember that adding a negative component is like moving in the opposite direction, but the method remains the same.

This question tests understanding of vector addition using the end-to-end method. The end-to-end (or head-to-tail) method of vector addition involves placing the tail of the second vector at the head of the first vector; the resultant vector then goes from the tail of the first vector to the head of the second vector. Starting with vector u = ⟨5, 0⟩ (5 units east) from the origin, we place vector v = ⟨0, 3⟩ (3 units north) so its tail is at the head of u, terminating at position ⟨5, 3⟩. The sum u + v is then the vector from the origin to ⟨5, 3⟩, which is ⟨5, 3⟩. Choice A is correct because it correctly describes the final position after walking 5 units east then 3 units north. Choice B incorrectly computes ⟨5 - 3, 3 - 3⟩ = ⟨2, 0⟩, suggesting confusion about how to combine perpendicular movements. For end-to-end addition, always place the tail (starting point) of the second vector at the head (endpoint) of the first vector, then draw the resultant from the original starting point to the final ending point.

This question tests understanding of vector addition using component-wise method and magnitude calculation. An important property of vector addition is that the magnitude of the sum is typically not equal to the sum of the magnitudes: |u + v| ≤ |u| + |v|, with equality only when the vectors point in exactly the same direction. For vectors u = ⟨3, 0⟩ and v = ⟨0, 4⟩, we compute u + v = ⟨3 + 0, 0 + 4⟩ = ⟨3, 4⟩. Computing |u + v| requires first finding the components of the sum ⟨3, 4⟩, then calculating √(3² + 4²) = √(9 + 16) = √25 = 5, which is less than |u| + |v| = 3 + 4 = 7. Choice B is correct because it properly applies the magnitude formula to the sum vector ⟨3, 4⟩. Choice A incorrectly assumes that |u + v| = |u| + |v|, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. Remember that |u + v| ≠ |u| + |v| in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (|u| + |v|) only when vectors point the same direction.

This question tests understanding of vector addition using the component-wise method with additive inverses. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨2, 3⟩ and v = ⟨-2, -3⟩, we compute u + v = ⟨2 + (-2), 3 + (-3)⟩ = ⟨0, 0⟩. Choice B is correct because additive inverse vectors have opposite components that cancel when added: 2 + (-2) = 0 and 3 + (-3) = 0, resulting in the zero vector. Choice A adds the absolute values instead of the signed values, computing ⟨|2| + |-2|, |3| + |-3|⟩ = ⟨4, 6⟩, which ignores the negative signs. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, and remember that additive inverses always sum to the zero vector ⟨0, 0⟩.

This question tests understanding of vector addition using the component-wise method with three vectors. Vectors are added component-wise: for multiple vectors, we add all corresponding components together separately. For vectors u = ⟨1, -2⟩, v = ⟨4, 3⟩, and w = ⟨-2, 1⟩, we compute u + v + w = ⟨1 + 4 + (-2), -2 + 3 + 1⟩ = ⟨3, 2⟩. Choice A is correct because it shows the component-wise calculation: the x-components give 1 + 4 - 2 = 3, and the y-components give -2 + 3 + 1 = 2. Choice B adds only the first two vectors correctly but makes an error with the third, computing ⟨1 + 4 + 2, -2 + 3 + 1⟩ = ⟨7, 2⟩, likely misreading the sign of w's first component. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—for multiple vectors, component-wise addition is often the most efficient approach.

This question tests understanding of vector addition using the component-wise method with negative components. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨-6, 2⟩ and v = ⟨1, -5⟩, we compute u + v = ⟨-6 + 1, 2 + (-5)⟩ = ⟨-5, -3⟩. Choice A is correct because it shows the component-wise calculation: the x-components give -6 + 1 = -5, and the y-components give 2 + (-5) = -3. Choice B subtracts the vectors instead of adding them, calculating ⟨-6 - 1, 2 - (-5)⟩ = ⟨-7, 7⟩, which gives u - v rather than u + v. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, being careful with negative signs.

This question tests understanding of vector addition using the parallelogram rule. The parallelogram rule for adding vectors involves placing both vectors tail-to-tail at a common initial point, completing a parallelogram using the vectors as adjacent sides, and taking the diagonal from the initial point as the sum. When both vectors u and v start from the same point, they form two adjacent sides of a parallelogram. The diagonal of this parallelogram starting from the common initial point represents u + v. Choice B is correct because it correctly describes the geometric construction of vector addition using the parallelogram rule. Choice C incorrectly assumes that |u + v| = |u| + |v|, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. The parallelogram rule shows why vector addition is commutative (u + v = v + u): both diagonals of the parallelogram represent the same sum, just constructed in different orders.

This question tests understanding of the relationship between vector magnitude and vector addition. An important property of vector addition is that the magnitude of the sum is typically not equal to the sum of the magnitudes: |u + v| ≤ |u| + |v|, with equality only when the vectors point in exactly the same direction. This inequality, known as the triangle inequality, states that the magnitude of a vector sum is always less than or equal to the sum of the individual magnitudes. Choice C is correct because it properly states the triangle inequality: |u + v| ≤ |u| + |v| for all vectors. Choice A incorrectly assumes that |u + v| = |u| + |v| for all vectors, but this equality only holds when vectors point in the same direction. Remember that |u + v| ≠ |u| + |v| in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (|u| + |v|) only when vectors point the same direction.

This question tests understanding of vector addition using component-wise addition in a physics context. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For the boat's velocity ⟨5, 0⟩ m/s and the current's velocity ⟨0, 2⟩ m/s, we compute the resultant velocity as ⟨5 + 0, 0 + 2⟩ = ⟨5, 2⟩ m/s. Choice A is correct because it properly applies the component-wise addition: the boat moves 5 m/s east relative to water, and the water moves 2 m/s north, giving a resultant velocity of ⟨5, 2⟩ m/s relative to the shore. Choice B incorrectly subtracts the y-component instead of adding it, calculating ⟨5, -2⟩, which would represent the boat moving against the current. Key to vector addition: when using component form, add x-components together and y-components together separately—this models how independent motions combine in different directions.

This question tests understanding of vector addition using the end-to-end method. The end-to-end (or head-to-tail) method of vector addition involves placing the tail of the second vector at the head of the first vector; the resultant vector then goes from the tail of the first vector to the head of the second vector. Starting with vector AB = ⟨2, 1⟩ from point A to point B, we place vector BC = ⟨3, 4⟩ so its tail is at B, terminating at point C; the sum AB + BC is then the vector AC from A to C, which is ⟨2 + 3, 1 + 4⟩ = ⟨5, 5⟩. Choice A is correct because it correctly describes the geometric construction in component form. Choice D adds only one component correctly but makes an arithmetic or sign error in the other component, possibly confusing the order. For end-to-end addition, always place the tail (starting point) of the second vector at the head (endpoint) of the first vector, then draw the resultant from the original starting point to the final ending point. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.