Subtract and Represent Vectors Graphically - Precalculus

1. For vectors $\vec v=\langle -1,4\rangle$ and $\vec w=\langle 3,-2\rangle$, compute $\vec v-\vec w$ by rewriting it as $\vec v+(-\vec w)$ and subtracting components.
   Answer: $\langle -4,6\rangle$ 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 = ⟨-1, 4⟩ and w = ⟨3, -2⟩, we compute v - w = ⟨-1 - 3, 4 - (-2)⟩ = ⟨-4, 6⟩. 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.
2. For vectors $\vec v = \langle 1, -3\rangle$ and $\vec w = \langle -4, 2\rangle$, use $\vec v-\vec w=\langle v_1-w_1,\, v_2-w_2\rangle$ (equivalently $\vec v+(-\vec w)$). What is $\vec v-\vec w$?
   Answer: $\langle 5, -5\rangle$ 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.
3. Let $\vec v = \langle 2, -4\rangle$ and $\vec w = \langle -3, 1\rangle$. Which statement correctly compares $\vec v-\vec w$ and $\vec w-\vec v$? (Use the idea that subtracting is adding the opposite.)
   Answer: $\vec v-\vec w = -\,(\vec w-\vec v)$ 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.
4. If $\vec w=\langle -3,2\rangle$, what is the opposite vector $-\vec w$ used in rewriting $\vec v-\vec w$ as $\vec v+(-\vec w)$?
   Answer: $\langle 3,-2\rangle$ This question tests understanding of vector subtraction and how to represent it as addition of the opposite. 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 = ⟨-3, 2⟩, we compute -w = ⟨-(-3), -2⟩ = ⟨3, -2⟩. Choice A is correct because it properly negates both components for -w, changing the sign of each component. Choice C 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). 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.
5. Using the vectors given, $\vec v=\langle 0, -4 \rangle$ and $\vec w=\langle -3, 2 \rangle$, what are the components of $\vec v-\vec w$? (Remember: subtracting $\vec w$ is the same as adding the opposite vector $-\vec w$.)
   Answer: $\langle 3, -6 \rangle$ 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 = ⟨0, -4⟩ and w = ⟨-3, 2⟩, we compute v - w = ⟨0 - (-3), -4 - 2⟩ = ⟨3, -6⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice A 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.
6. In the coordinate plane shown, vectors $\vec{AB}$ and $\vec{CD}$ are drawn. To construct $\vec{AB} - \vec{CD}$ graphically using the tip-to-tail method, which sequence of steps is correct?
   Answer: Place the tail of $-\vec{CD}$ at the tip of $\vec{AB}$, then draw from the tail of $\vec{AB}$ to the tip of $-\vec{CD}$ Vector subtraction $\vec{AB} - \vec{CD}$ is equivalent to $\vec{AB} + (-\vec{CD})$. The correct graphical method places the tail of the additive inverse $-\vec{CD}$ at the tip of $\vec{AB}$, then draws the resultant from the tail of $\vec{AB}$ to the tip of $-\vec{CD}$. Choice A describes vector addition, not subtraction. Choice C reverses the construction order. Choice D reverses the roles of the vectors.
7. Using the vectors given, what is $-\vec w$ if $\vec w=\langle -7,3\rangle$? (This is the opposite vector used to rewrite $\vec v-\vec w$ as $\vec v+(-\vec w)$.)
   Answer: $\langle 7,-3\rangle$ 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⟩. If w = ⟨-7, 3⟩, then -w = ⟨-(-7), -3⟩ = ⟨7, -3⟩, which means we negate both components; this gives a vector with the same length but pointing in the exact opposite direction. Choice C is correct because it correctly negates both components for -w. Choice B 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). 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.
8. Compare $\vec v-\vec w$ and $\vec w-\vec v$ for $\vec v=\langle 2,1\rangle$ and $\vec w=\langle 5,3\rangle$. Which statement is true? (Think of subtraction as adding the opposite.)
   Answer: $\vec v-\vec w=-(\vec w-\vec v)$. 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⟩. Computing both: v - w = ⟨2 - 5, 1 - 3⟩ = ⟨-3, -2⟩ and w - v = ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩, 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. 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. 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.
9. An object's displacement is $\vec v=\langle 3,-2\rangle$ meters, and a wind effect is modeled by $\vec w=\langle -1,4\rangle$ meters. The adjusted displacement is $\vec v-\vec w$ (i.e., $\vec v+(-\vec w)$). What are the components of $\vec v-\vec w$?
   Answer: $\langle 4,-6\rangle$ 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 = ⟨3, -2⟩ and w = ⟨-1, 4⟩, we compute v - w = ⟨3 - (-1), -2 - 4⟩ = ⟨4, -6⟩. 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.
10. For vectors $\vec v=\langle -2,6\rangle$ and $\vec w=\langle 3,-1\rangle$, rewrite $\vec v-\vec w$ as $\vec v+(-\vec w)$ and find the components of $\vec v-\vec w$.
    Answer: $\langle -5,7\rangle$ 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 = ⟨-2, 6⟩ and w = ⟨3, -1⟩, we compute v - w = ⟨-2 - 3, 6 - (-1)⟩ = ⟨-5, 7⟩. 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.
11. Given vectors $\vec{x} = \langle -2, 6 \rangle$ and $\vec{y} = \langle 4, -1 \rangle$, if $\vec{z} = 2\vec{x} - 3\vec{y}$, what are the components of $\vec{z}$?
    Answer: $\langle -16, 15 \rangle$ When you encounter vector operations like scalar multiplication and vector addition/subtraction, you need to apply these operations component-wise to each coordinate. To find $\vec{z} = 2\vec{x} - 3\vec{y}$, first calculate each scalar multiplication separately, then subtract the results. Start with $2\vec{x} = 2\langle -2, 6 \rangle = \langle 2(-2), 2(6) \rangle = \langle -4, 12 \rangle$. Next, find $3\vec{y} = 3\langle 4, -1 \rangle = \langle 3(4), 3(-1) \rangle = \langle 12, -3 \rangle$. Now subtract: $\vec{z} = \langle -4, 12 \rangle - \langle 12, -3 \rangle = \langle -4 - 12, 12 - (-3) \rangle = \langle -16, 15 \rangle$. This confirms answer C is correct. Let's examine why the other answers are wrong. Answer A $\langle 8, 9 \rangle$ likely comes from incorrectly adding instead of subtracting, or mixing up the scalar multipliers. Answer B $\langle -10, 9 \rangle$ suggests a calculation error in the first component, possibly from $-4 - 6 = -10$ instead of the correct $-4 - 12 = -16$. Answer D $\langle -4, 18 \rangle$ appears to result from forgetting to multiply $\vec{y}$ by 3 before subtracting, giving $\langle -4, 12 \rangle - \langle 4, -1 \rangle = \langle -8, 13 \rangle$, which still doesn't match. Remember: always perform scalar multiplication first, then handle vector addition or subtraction component by component. Double-check your arithmetic, especially with negative numbers, as sign errors are common in these problems.
12. Vector $\vec v = \langle 5, 3\rangle$ and vector $\vec w = \langle 2, 1\rangle$. Using vector subtraction as adding the opposite (so $\vec v-\vec w=\vec v+(-\vec w)$), what are the components of $\vec v-\vec w$?
    Answer: $\langle 3, 2\rangle$ 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 = ⟨5, 3⟩ and w = ⟨2, 1⟩, we first find -w = ⟨-2, -1⟩, then compute v - w = v + (-w) = ⟨5, 3⟩ + ⟨-2, -1⟩ = ⟨5 + (-2), 3 + (-1)⟩ = ⟨3, 2⟩. Choice B is correct because it applies the component subtraction formula correctly, giving ⟨3, 2⟩. Choice A adds the vectors instead of subtracting them, calculating v + w = ⟨7, 4⟩ when we need v - w. 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.
13. On a coordinate plane, $\vec v$ and $\vec w$ are drawn tail-to-tail at the origin with tips at $V(2,-1)$ and $W(-1,1)$, respectively. Using the tip-to-tip method (vector from the tip of $\vec w$ to the tip of $\vec v$), what is $\vec v-\vec w$ in component form?
    Answer: $\langle 3,-2\rangle$ 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. When v and w are drawn with their tails at the same point, visualize walking from the tip of w to the tip of v—that path represents v - w. This is equivalent to the triangle rule: v, w, and v - w form a closed triangle. Choice C is correct because it properly describes the tip-to-tip method. Choice B reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. 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). 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.
14. For vectors $\vec v=\langle 5,3\rangle$ and $\vec w=\langle 2,1\rangle$, what are the components of $\vec v-\vec w$? (Recall: vector subtraction means adding the opposite, $\vec v-\vec w=\vec v+(-\vec w)$.)
    Answer: $\langle 3,2\rangle$ 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 = ⟨5, 3⟩ and w = ⟨2, 1⟩, we compute v - w = ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩. Choice B is correct because it applies the component subtraction formula correctly, giving ⟨3, 2⟩. Choice A adds the vectors instead of subtracting them, calculating v + w = ⟨5 + 2, 3 + 1⟩ = ⟨7, 4⟩ when we need v - w. 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.
15. Compare $\vec v-\vec w$ and $\vec w-\vec v$ for $\vec v=\langle -1, 4 \rangle$ and $\vec w=\langle 2, 0 \rangle$. Which statement is true?
    Answer: $\vec v-\vec w=-(\vec w-\vec v)$. 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⟩. Computing both: v - w = ⟨-1 - 2, 4 - 0⟩ = ⟨-3, 4⟩ and w - v = ⟨2 - (-1), 0 - 4⟩ = ⟨3, -4⟩, 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. 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. 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.
16. Vectors $\vec v$ and $\vec w$ are drawn tail-to-tail from the same initial point. How is $\vec v-\vec w$ represented graphically using the tip-to-tip method (consistent with $\vec v-\vec w=\vec v+(-\vec w)$)?
    Answer: The vector from the tip of $\vec w$ to the tip of $\vec v$. This question tests understanding of vector subtraction and how to represent it graphically using the tip-to-tip method. 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⟩. 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. Choice C is correct because it properly describes the tip-to-tip method. 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). 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.
17. Let $\vec v = \langle -1, 4\rangle$ and $\vec w = \langle 3, -2\rangle$. Since $\vec v-\vec w=\vec v+(-\vec w)$ and $-\vec w=\langle -w_1, -w_2\rangle$, what is $\vec v-\vec w$ in component form?
    Answer: $\langle -4, 6\rangle$ 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.
18. Vectors $\vec v$ and $\vec w$ are drawn tail-to-tail from the origin on a coordinate plane, with $\vec v=\langle 4,1\rangle$ and $\vec w=\langle 1,3\rangle$. Using the tip-to-tip method, what are the components of $\vec v-\vec w$?
    Answer: $\langle 3,-2\rangle$ 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. When v and w are drawn with their tails at the same point, visualize walking from the tip of w to the tip of v—that path represents v - w. This is equivalent to the triangle rule: v, w, and v - w form a closed triangle. Choice A is correct because it properly describes the tip-to-tip method. Choice C reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. 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). 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.
19. Let $\vec v=\langle 2,1\rangle$ and $\vec w=\langle 5,3\rangle$. Since $\vec v-\vec w=\vec v+(-\vec w)$, what is $\vec v-\vec w$ in component form?
    Answer: $\langle -3,-2\rangle$ 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 = ⟨2, 1⟩ and w = ⟨5, 3⟩, we compute v - w = ⟨2 - 5, 1 - 3⟩ = ⟨-3, -2⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice A 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. 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.
20. For vectors $\vec v = \langle 5, 3 \rangle$ and $\vec w = \langle 2, 1 \rangle$, use component-wise subtraction (equivalently $\vec v-\vec w=\vec v+(-\vec w)$) to find $\vec v-\vec w$ in $\langle a,b\rangle$ form.
    Answer: $\langle 3, 2 \rangle$ 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 = ⟨5, 3⟩ and w = ⟨2, 1⟩, we compute v - w = ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice A adds the vectors instead of subtracting them, calculating v + w = ⟨5 + 2, 3 + 1⟩ = ⟨7, 4⟩ when we need v - w. 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.

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