This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨4, 0⟩ and w = ⟨1, 0⟩, we compute v - w = ⟨4 - 1, 0 - 0⟩ = ⟨3, 0⟩. Choice A is correct because it applies the component subtraction formula correctly, giving ⟨3, 0⟩. Choice B adds the vectors instead of subtracting them, calculating v + w = ⟨5, 0⟩ when we need v - w. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v: ⟨1, 0⟩ + ⟨3, 0⟩ = ⟨4, 0⟩ ✓.

This question tests understanding of vector subtraction and how to represent it as addition of the opposite. To subtract vectors component-wise, subtract the corresponding components: if v = ⟨v₁, v₂⟩ and w = ⟨w₁, w₂⟩, then v - w = ⟨v₁ - w₁, v₂ - w₂⟩. For v = ⟨-1, 4⟩ and w = ⟨3, -2⟩, we compute v - w = ⟨-1 - 3, 4 - (-2)⟩ = ⟨-4, 6⟩. Choice A is correct because it applies the component subtraction formula correctly, giving ⟨-4, 6⟩. Choice D makes an arithmetic error in the subtraction, calculating ⟨-4, 2⟩ instead of ⟨-4, 6⟩ by incorrectly computing 4 - (-2) = 2 rather than 6. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition.

This question tests understanding of vector subtraction and how to represent it using component-wise subtraction. To subtract vectors component-wise, subtract the corresponding components: if v = ⟨v₁, v₂⟩ and w = ⟨w₁, w₂⟩, then v - w = ⟨v₁ - w₁, v₂ - w₂⟩. For v = ⟨2, 1⟩ and w = ⟨5, 3⟩, we compute v - w = ⟨2 - 5, 1 - 3⟩ = ⟨-3, -2⟩. Choice C is correct because it applies the component subtraction formula correctly, giving ⟨-3, -2⟩. Choice B reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped: ⟨3, 2⟩. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition.

This question tests understanding of vector subtraction and how to represent it as addition of the opposite, specifically finding the opposite vector. If w = ⟨w₁, w₂⟩, then -w = ⟨-w₁, -w₂⟩, which means we negate both components; this gives a vector with the same length but pointing in the exact opposite direction. For w = ⟨-2, 3⟩, we compute -w = ⟨-(-2), -3⟩ = ⟨2, -3⟩. Choice B is correct because it properly negates both components for -w, giving ⟨2, -3⟩. Choice A only negates one component when finding -w, but the opposite of a vector requires negating both components. When finding the opposite of a vector, -w = ⟨-w₁, -w₂⟩, negate every component; this creates a vector with identical magnitude but pointing in the exactly opposite direction (180° rotation).

This question tests understanding of vector subtraction and how to represent it using the component-wise formula. To subtract vectors component-wise, subtract the corresponding components: if v = ⟨v₁, v₂⟩ and w = ⟨w₁, w₂⟩, then v - w = ⟨v₁ - w₁, v₂ - w₂⟩. For v = ⟨1, -3⟩ and w = ⟨-4, 2⟩, we compute v - w = ⟨1 - (-4), -3 - 2⟩ = ⟨5, -5⟩. Choice C is correct because it applies the component subtraction formula correctly, giving ⟨5, -5⟩. Choice B reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped: ⟨-5, 5⟩. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result.

This question tests understanding of vector subtraction and the non-commutative nature of this operation. To subtract vectors component-wise, subtract the corresponding components: if v = ⟨v₁, v₂⟩ and w = ⟨w₁, w₂⟩, then w - v = ⟨w₁ - v₁, w₂ - v₂⟩. For v = ⟨3, 2⟩ and w = ⟨-1, 5⟩, we compute w - v = ⟨-1 - 3, 5 - 2⟩ = ⟨-4, 3⟩. Choice B is correct because it applies the component subtraction formula correctly, giving ⟨-4, 3⟩. Choice A computes v - w instead of w - v, which gives ⟨4, -3⟩, the opposite vector with all signs flipped. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition.

This question tests understanding of the non-commutative nature of vector subtraction. Vector subtraction is not commutative: v - w ≠ w - v in general; in fact, v - w = -(w - v), meaning they point in opposite directions with the same magnitude. Computing both: v - w = ⟨2 - (-3), -4 - 1⟩ = ⟨5, -5⟩ and w - v = ⟨-3 - 2, 1 - (-4)⟩ = ⟨-5, 5⟩, we see that w - v = -(v - w), confirming they have equal magnitudes but opposite directions. Choice B is correct because it correctly identifies the relationship v - w = -(w - v). Choice A incorrectly assumes vector subtraction is commutative, claiming v - w = w - v, when in fact they are opposites. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition.

This question tests understanding of vector subtraction and how to represent it graphically using the tip-to-tip method. Graphically, when vectors v and w are drawn from the same initial point (tail-to-tail), the vector v - w is represented by the vector from the tip of w to the tip of v, forming a triangle. To find v - w graphically, we can rewrite it as v + (-w): first draw v, then draw -w (the opposite of w), place the tail of -w at the head of v, and the resultant from the tail of v to the head of -w is v - w. Choice C is correct because it properly describes the tip-to-tip method, going from the tip of w to the tip of v. Choice A describes the graphical vector going from the tip of v to the tip of w, but v - w goes from the tip of w to the tip of v (the opposite direction). For the tip-to-tip graphical method, place both vectors tail-to-tail and draw the difference vector from the tip of what you're subtracting (w) to the tip of what you're subtracting from (v).

This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨0, -3⟩ and w = ⟨-2, 4⟩, we compute v - w = ⟨0 - (-2), -3 - 4⟩ = ⟨2, -7⟩. Choice B is correct because it applies the component subtraction formula correctly, properly handling the double negative in the first component. Choice C makes an arithmetic error in the subtraction, calculating ⟨-2, -7⟩ instead of ⟨2, -7⟩, likely by incorrectly handling the subtraction of the negative first component. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨6, 0⟩ and w = ⟨1, 0⟩, we compute v - w = ⟨6 - 1, 0 - 0⟩ = ⟨5, 0⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice C reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.