Subtract and Represent Vectors Graphically - Pre-Calculus
Card 1 of 30
Which vector should you add to $\mathbf{v}$ to compute $\mathbf{v}-\mathbf{w}$ graphically?
Which vector should you add to $\mathbf{v}$ to compute $\mathbf{v}-\mathbf{w}$ graphically?
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$-\mathbf{w}$. Use the additive inverse to convert subtraction to addition.
$-\mathbf{w}$. Use the additive inverse to convert subtraction to addition.
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What is the definition of vector subtraction $\mathbf{v}-\mathbf{w}$ in terms of addition?
What is the definition of vector subtraction $\mathbf{v}-\mathbf{w}$ in terms of addition?
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$\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction is defined as adding the additive inverse.
$\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction is defined as adding the additive inverse.
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What is the identity that rewrites vector subtraction $\mathbf{v}-\mathbf{w}$ using addition?
What is the identity that rewrites vector subtraction $\mathbf{v}-\mathbf{w}$ using addition?
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$\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction becomes addition of the additive inverse.
$\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction becomes addition of the additive inverse.
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Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle 4,1\rangle$ and $\mathbf{w}=\langle -2,5\rangle$.
Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle 4,1\rangle$ and $\mathbf{w}=\langle -2,5\rangle$.
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$\langle 6,-4\rangle$. $\langle 4-(-2), 1-5\rangle = \langle 4+2, -4\rangle$.
$\langle 6,-4\rangle$. $\langle 4-(-2), 1-5\rangle = \langle 4+2, -4\rangle$.
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What is the additive inverse of a vector $\mathbf{w}$ written in component form?
What is the additive inverse of a vector $\mathbf{w}$ written in component form?
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If $\mathbf{w}=\langle a,b\rangle$, then $-\mathbf{w}=\langle -a,-b\rangle$. Negate each component to get the additive inverse.
If $\mathbf{w}=\langle a,b\rangle$, then $-\mathbf{w}=\langle -a,-b\rangle$. Negate each component to get the additive inverse.
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What is the magnitude relationship between $\mathbf{w}$ and $-\mathbf{w}$?
What is the magnitude relationship between $\mathbf{w}$ and $-\mathbf{w}$?
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$\lVert -\mathbf{w}\rVert=\lVert \mathbf{w}\rVert$. Negating a vector preserves its magnitude.
$\lVert -\mathbf{w}\rVert=\lVert \mathbf{w}\rVert$. Negating a vector preserves its magnitude.
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What is the direction relationship between $\mathbf{w}$ and $-\mathbf{w}$?
What is the direction relationship between $\mathbf{w}$ and $-\mathbf{w}$?
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$-\mathbf{w}$ points opposite $\mathbf{w}$. Negating reverses the vector's direction by 180°.
$-\mathbf{w}$ points opposite $\mathbf{w}$. Negating reverses the vector's direction by 180°.
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State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2\rangle$ and $\mathbf{w}=\langle w_1,w_2\rangle$.
State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2\rangle$ and $\mathbf{w}=\langle w_1,w_2\rangle$.
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$\mathbf{v}-\mathbf{w}=\langle v_1-w_1,,v_2-w_2\rangle$. Subtract corresponding components.
$\mathbf{v}-\mathbf{w}=\langle v_1-w_1,,v_2-w_2\rangle$. Subtract corresponding components.
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State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2,v_3\rangle$ and $\mathbf{w}=\langle w_1,w_2,w_3\rangle$.
State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2,v_3\rangle$ and $\mathbf{w}=\langle w_1,w_2,w_3\rangle$.
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$\mathbf{v}-\mathbf{w}=\langle v_1-w_1,,v_2-w_2,,v_3-w_3\rangle$. Subtract each corresponding component in 3D.
$\mathbf{v}-\mathbf{w}=\langle v_1-w_1,,v_2-w_2,,v_3-w_3\rangle$. Subtract each corresponding component in 3D.
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In a tip-to-tail diagram, how do you place $-\mathbf{w}$ to represent $\mathbf{v}-\mathbf{w}$?
In a tip-to-tail diagram, how do you place $-\mathbf{w}$ to represent $\mathbf{v}-\mathbf{w}$?
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Place tail of $-\mathbf{w}$ at tip of $\mathbf{v}$. Standard tip-to-tail addition method.
Place tail of $-\mathbf{w}$ at tip of $\mathbf{v}$. Standard tip-to-tail addition method.
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In a tip-to-tail diagram for $\mathbf{v}+(-\mathbf{w})$, what does the resultant connect?
In a tip-to-tail diagram for $\mathbf{v}+(-\mathbf{w})$, what does the resultant connect?
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From tail of $\mathbf{v}$ to tip of $-\mathbf{w}$. The resultant spans from start to end of the path.
From tail of $\mathbf{v}$ to tip of $-\mathbf{w}$. The resultant spans from start to end of the path.
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When $\mathbf{v}$ and $\mathbf{w}$ share the same tail, what geometric vector equals $\mathbf{v}-\mathbf{w}$?
When $\mathbf{v}$ and $\mathbf{w}$ share the same tail, what geometric vector equals $\mathbf{v}-\mathbf{w}$?
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The vector from tip of $\mathbf{w}$ to tip of $\mathbf{v}$. Direct path from $\mathbf{w}$'s endpoint to $\mathbf{v}$'s endpoint.
The vector from tip of $\mathbf{w}$ to tip of $\mathbf{v}$. Direct path from $\mathbf{w}$'s endpoint to $\mathbf{v}$'s endpoint.
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Find $-\mathbf{w}$ if $\mathbf{w}=\langle 7,-3\rangle$.
Find $-\mathbf{w}$ if $\mathbf{w}=\langle 7,-3\rangle$.
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$\langle -7,3\rangle$. Negate each component: $7$ becomes $-7$, $-3$ becomes $3$.
$\langle -7,3\rangle$. Negate each component: $7$ becomes $-7$, $-3$ becomes $3$.
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Which expression is equivalent to $\mathbf{w}-\mathbf{v}$ written using $\mathbf{v}-\mathbf{w}$?
Which expression is equivalent to $\mathbf{w}-\mathbf{v}$ written using $\mathbf{v}-\mathbf{w}$?
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$\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.
$\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.
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Find and correct the error: A student wrote $\mathbf{v}-\mathbf{w}=\langle v_1-w_2,,v_2-w_1\rangle$.
Find and correct the error: A student wrote $\mathbf{v}-\mathbf{w}=\langle v_1-w_2,,v_2-w_1\rangle$.
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Correct: $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,,v_2-w_2\rangle$. Student mixed indices; subtract matching components.
Correct: $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,,v_2-w_2\rangle$. Student mixed indices; subtract matching components.
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If $\mathbf{v}=\langle 2,1\rangle$ and $\mathbf{w}=\langle -4,3\rangle$, what is $\mathbf{v}-\mathbf{w}$?
If $\mathbf{v}=\langle 2,1\rangle$ and $\mathbf{w}=\langle -4,3\rangle$, what is $\mathbf{v}-\mathbf{w}$?
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$\langle 6,-2\rangle$. Subtract components: $2-(-4)=6$ and $1-3=-2$.
$\langle 6,-2\rangle$. Subtract components: $2-(-4)=6$ and $1-3=-2$.
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What is the additive inverse of a vector $\mathbf{w}$ called and written as?
What is the additive inverse of a vector $\mathbf{w}$ called and written as?
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Additive inverse: $-\mathbf{w}$. The negative of a vector is its additive inverse.
Additive inverse: $-\mathbf{w}$. The negative of a vector is its additive inverse.
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What are the magnitude and direction of $-\mathbf{w}$ compared with $\mathbf{w}$?
What are the magnitude and direction of $-\mathbf{w}$ compared with $\mathbf{w}$?
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Same magnitude, opposite direction. Negating a vector reverses its direction but preserves its length.
Same magnitude, opposite direction. Negating a vector reverses its direction but preserves its length.
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What is the component form of $-\langle a,b\rangle$?
What is the component form of $-\langle a,b\rangle$?
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$-\langle a,b\rangle=\langle -a,-b\rangle$. Negate each component to find the additive inverse.
$-\langle a,b\rangle=\langle -a,-b\rangle$. Negate each component to find the additive inverse.
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What is the component-wise subtraction rule for $\langle a,b\rangle-\langle c,d\rangle$?
What is the component-wise subtraction rule for $\langle a,b\rangle-\langle c,d\rangle$?
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$\langle a,b\rangle-\langle c,d\rangle=\langle a-c,,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.
$\langle a,b\rangle-\langle c,d\rangle=\langle a-c,,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.
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What is $\langle 7,-2\rangle-\langle 3,5\rangle$?
What is $\langle 7,-2\rangle-\langle 3,5\rangle$?
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$\langle 4,-7\rangle$. Apply component-wise subtraction: $7-3=4$ and $-2-5=-7$.
$\langle 4,-7\rangle$. Apply component-wise subtraction: $7-3=4$ and $-2-5=-7$.
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What is $\langle -1,4\rangle-\langle 6,-3\rangle$?
What is $\langle -1,4\rangle-\langle 6,-3\rangle$?
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$\langle -7,7\rangle$. Apply component-wise subtraction: $-1-6=-7$ and $4-(-3)=7$.
$\langle -7,7\rangle$. Apply component-wise subtraction: $-1-6=-7$ and $4-(-3)=7$.
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What is $-\langle -8,2\rangle$?
What is $-\langle -8,2\rangle$?
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$\langle 8,-2\rangle$. Negate each component: $-(-8)=8$ and $-(2)=-2$.
$\langle 8,-2\rangle$. Negate each component: $-(-8)=8$ and $-(2)=-2$.
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What is $\mathbf{v}-\mathbf{0}$ for any vector $\mathbf{v}$?
What is $\mathbf{v}-\mathbf{0}$ for any vector $\mathbf{v}$?
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$\mathbf{v}$. Subtracting the zero vector leaves any vector unchanged.
$\mathbf{v}$. Subtracting the zero vector leaves any vector unchanged.
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What is $\mathbf{0}-\mathbf{w}$ for any vector $\mathbf{w}$?
What is $\mathbf{0}-\mathbf{w}$ for any vector $\mathbf{w}$?
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$-\mathbf{w}$. Zero minus a vector equals the negative of that vector.
$-\mathbf{w}$. Zero minus a vector equals the negative of that vector.
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What is $\mathbf{v}-\mathbf{v}$ for any vector $\mathbf{v}$?
What is $\mathbf{v}-\mathbf{v}$ for any vector $\mathbf{v}$?
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$\mathbf{0}$. Any vector minus itself equals the zero vector.
$\mathbf{0}$. Any vector minus itself equals the zero vector.
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Which vector equals $\mathbf{w}-\mathbf{v}$ in terms of $\mathbf{v}-\mathbf{w}$?
Which vector equals $\mathbf{w}-\mathbf{v}$ in terms of $\mathbf{v}-\mathbf{w}$?
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$\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.
$\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.
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Identify the vector represented by the arrow from the tip of $\mathbf{w}$ to the tip of $\mathbf{v}$ (tails together).
Identify the vector represented by the arrow from the tip of $\mathbf{w}$ to the tip of $\mathbf{v}$ (tails together).
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$\mathbf{v}-\mathbf{w}$. The difference vector points from $\mathbf{w}$'s tip to $\mathbf{v}$'s tip.
$\mathbf{v}-\mathbf{w}$. The difference vector points from $\mathbf{w}$'s tip to $\mathbf{v}$'s tip.
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If $\mathbf{v}=\langle -5,0\rangle$ and $\mathbf{w}=\langle 1,-7\rangle$, what is $\mathbf{v}-\mathbf{w}$?
If $\mathbf{v}=\langle -5,0\rangle$ and $\mathbf{w}=\langle 1,-7\rangle$, what is $\mathbf{v}-\mathbf{w}$?
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$\langle -6,7\rangle$. Subtract components: $-5-1=-6$ and $0-(-7)=7$.
$\langle -6,7\rangle$. Subtract components: $-5-1=-6$ and $0-(-7)=7$.
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What is the vector from point $A(1,2)$ to point $B(6,-1)$ written as $\overrightarrow{AB}$?
What is the vector from point $A(1,2)$ to point $B(6,-1)$ written as $\overrightarrow{AB}$?
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$\overrightarrow{AB}=\langle 5,-3\rangle$. Vector from $A$ to $B$ is $B-A$: $(6-1, -1-2)$.
$\overrightarrow{AB}=\langle 5,-3\rangle$. Vector from $A$ to $B$ is $B-A$: $(6-1, -1-2)$.
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