Precalculus Flashcards: Subtract And Represent Vectors Graphically

What this deck covers

This deck focuses on Subtract And Represent Vectors Graphically, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $\mathbf{v}-\mathbf{0}$ for any vector $\mathbf{v}$?

Answer: $\mathbf{v}$. Subtracting the zero vector leaves any vector unchanged.

Flashcard 2: If $\mathbf{v}=\langle 2,1\rangle$ and $\mathbf{w}=\langle -4,3\rangle$, what is $\mathbf{v}-\mathbf{w}$?

Answer: $\langle 6,-2\rangle$. Subtract components: $2-(-4)=6$ and $1-3=-2$.

Flashcard 3: What is $\overrightarrow{PQ}$ if $P(-3,4)$ and $Q(2,9)$?

Answer: $\overrightarrow{PQ}=\langle 5,5\rangle$. Vector from $P$ to $Q$ is $Q-P$: $(2-(-3), 9-4)$.

Flashcard 4: In a tip-to-tail diagram for $\mathbf{v}+(-\mathbf{w})$, what does the resultant connect?

Answer: From tail of $\mathbf{v}$ to tip of $-\mathbf{w}$. The resultant spans from start to end of the path.

Flashcard 5: If $\mathbf{v}=\langle -5,0\rangle$ and $\mathbf{w}=\langle 1,-7\rangle$, what is $\mathbf{v}-\mathbf{w}$?

Answer: $\langle -6,7\rangle$. Subtract components: $-5-1=-6$ and $0-(-7)=7$.

Flashcard 6: Which vector equals $\mathbf{w}-\mathbf{v}$ in terms of $\mathbf{v}-\mathbf{w}$?

Answer: $\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.

Flashcard 7: State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2,v_3\rangle$ and $\mathbf{w}=\langle w_1,w_2,w_3\rangle$.

Answer: $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,\,v_2-w_2,\,v_3-w_3\rangle$. Subtract each corresponding component in 3D.

Flashcard 8: What is the identity that rewrites vector subtraction $\mathbf{v}-\mathbf{w}$ using addition?

Answer: $\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction becomes addition of the additive inverse.

Flashcard 9: What is $\langle -1,4\rangle-\langle 6,-3\rangle$?

Answer: $\langle -7,7\rangle$. Apply component-wise subtraction: $-1-6=-7$ and $4-(-3)=7$.

Flashcard 10: What is $\mathbf{0}-\mathbf{w}$ for any vector $\mathbf{w}$?

Answer: $-\mathbf{w}$. Zero minus any vector equals its negative.

Flashcard 11: Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle -3,8\rangle$ and $\mathbf{w}=\langle 1,-6\rangle$.

Answer: $\langle -4,14\rangle$. $\langle -3-1, 8-(-6)\rangle = \langle -4, 8+6\rangle$.

Flashcard 12: Identify the correct simplification of $\mathbf{v}-(-\mathbf{w})$.

Answer: $\mathbf{v}+\mathbf{w}$. Double negative becomes positive: $-(-\mathbf{w}) = \mathbf{w}$.

Flashcard 13: Find $\mathbf{v}$ if $\mathbf{v}-\langle 3,-2\rangle=\langle -1,5\rangle$.

Answer: $\mathbf{v}=\langle 2,3\rangle$. Add $\langle 3,-2\rangle$ to both sides: $\mathbf{v}=\langle -1,5\rangle+\langle 3,-2\rangle$.

Flashcard 14: What is $\mathbf{v}-\mathbf{v}$ for any vector $\mathbf{v}$?

Answer: $\mathbf{0}$. Any vector minus itself equals the zero vector.

Flashcard 15: What is the definition of vector subtraction $\mathbf{v}-\mathbf{w}$ in terms of addition?

Answer: $\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction is defined as adding the additive inverse.

Flashcard 16: Which expression is equivalent to $\mathbf{w}-\mathbf{v}$ written using $\mathbf{v}-\mathbf{w}$?

Answer: $\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.

Flashcard 17: Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle 4,1\rangle$ and $\mathbf{w}=\langle -2,5\rangle$.

Answer: $\langle 6,-4\rangle$. $\langle 4-(-2), 1-5\rangle = \langle 4+2, -4\rangle$.

Flashcard 18: What is the magnitude relationship between $\mathbf{w}$ and $-\mathbf{w}$?

Answer: $\lVert -\mathbf{w}\rVert=\lVert \mathbf{w}\rVert$. Negating a vector preserves its magnitude.

Flashcard 19: What is $\mathbf{v}-\mathbf{0}$ for any vector $\mathbf{v}$?

Answer: $\mathbf{v}$. Subtracting the zero vector leaves any vector unchanged.

Flashcard 20: When $\mathbf{v}$ and $\mathbf{w}$ share the same tail, what geometric vector equals $\mathbf{v}-\mathbf{w}$?

Answer: The vector from tip of $\mathbf{w}$ to tip of $\mathbf{v}$. Direct path from $\mathbf{w}$'s endpoint to $\mathbf{v}$'s endpoint.

Flashcard 21: State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2\rangle$ and $\mathbf{w}=\langle w_1,w_2\rangle$.

Answer: $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,\,v_2-w_2\rangle$. Subtract corresponding components.

Flashcard 22: Find and correct the error: $\langle a,b\rangle-\langle c,d\rangle=\langle a+c,\,b+d\rangle$.

Answer: Correct: $\langle a-c,\,b-d\rangle$. The error shows addition; subtraction requires $a-c$ and $b-d$.

Flashcard 23: What is $-\langle -8,2\rangle$?

Answer: $\langle 8,-2\rangle$. Negate each component: $-(-8)=8$ and $-(2)=-2$.

Flashcard 24: What is the component-wise subtraction rule for $\langle a,b\rangle-\langle c,d\rangle$?

Answer: $\langle a,b\rangle-\langle c,d\rangle=\langle a-c,\,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.

Flashcard 25: Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle 2,-1,5\rangle$ and $\mathbf{w}=\langle -4,3,2\rangle$.

Answer: $\langle 6,-4,3\rangle$. $\langle 2-(-4), -1-3, 5-2\rangle = \langle 6, -4, 3\rangle$.

Flashcard 26: Which vector should you add to $\mathbf{v}$ to compute $\mathbf{v}-\mathbf{w}$ graphically?

Answer: $-\mathbf{w}$. Use the additive inverse to convert subtraction to addition.

Flashcard 27: What is $\langle 7,-2\rangle-\langle 3,5\rangle$?

Answer: $\langle 4,-7\rangle$. Apply component-wise subtraction: $7-3=4$ and $-2-5=-7$.

Flashcard 28: Find $-\mathbf{w}$ if $\mathbf{w}=\langle 7,-3\rangle$.

Answer: $\langle -7,3\rangle$. Negate each component: $7$ becomes $-7$, $-3$ becomes $3$.

Flashcard 29: What are the magnitude and direction of $-\mathbf{w}$ compared with $\mathbf{w}$?

Answer: Same magnitude, opposite direction. Negating a vector reverses its direction but preserves its length.

Flashcard 30: What is the vector from point $A(1,2)$ to point $B(6,-1)$ written as $\overrightarrow{AB}$?

Answer: $\overrightarrow{AB}=\langle 5,-3\rangle$. Vector from $A$ to $B$ is $B-A$: $(6-1, -1-2)$.

Flashcard 31: Find and correct the error: A student wrote $\mathbf{v}-\mathbf{w}=\langle v_1-w_2,\,v_2-w_1\rangle$.

Answer: Correct: $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,\,v_2-w_2\rangle$. Student mixed indices; subtract matching components.

Flashcard 32: What is the component form of $-\langle a,b\rangle$?

Answer: $-\langle a,b\rangle=\langle -a,-b\rangle$. Negate each component to find the additive inverse.

Flashcard 33: What is the direction relationship between $\mathbf{w}$ and $-\mathbf{w}$?

Answer: $-\mathbf{w}$ points opposite $\mathbf{w}$. Negating reverses the vector's direction by 180°.

Flashcard 34: What is the additive inverse of a vector $\mathbf{w}$ called and written as?

Answer: Additive inverse: $-\mathbf{w}$. The negative of a vector is its additive inverse.

Flashcard 35: What is $\mathbf{v}-\mathbf{v}$ for any vector $\mathbf{v}$?

Answer: $\mathbf{0}$. Any vector minus itself equals the zero vector.

Flashcard 36: What is the additive inverse of a vector $\mathbf{w}$ written in component form?

Answer: If $\mathbf{w}=\langle a,b\rangle$, then $-\mathbf{w}=\langle -a,-b\rangle$. Negate each component to get the additive inverse.

Flashcard 37: What is $\mathbf{0}-\mathbf{w}$ for any vector $\mathbf{w}$?

Answer: $-\mathbf{w}$. Zero minus a vector equals the negative of that vector.

Flashcard 38: Identify the vector represented by the arrow from the tip of $\mathbf{w}$ to the tip of $\mathbf{v}$ (tails together).

Answer: $\mathbf{v}-\mathbf{w}$. The difference vector points from $\mathbf{w}$'s tip to $\mathbf{v}$'s tip.

Flashcard 39: In a tip-to-tail diagram, how do you place $-\mathbf{w}$ to represent $\mathbf{v}-\mathbf{w}$?

Answer: Place tail of $-\mathbf{w}$ at tip of $\mathbf{v}$. Standard tip-to-tail addition method.