Add Vectors in Different Ways - Pre-Calculus
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What is the parallelogram rule statement for $\vec{u}+\vec{v}$?
What is the parallelogram rule statement for $\vec{u}+\vec{v}$?
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Diagonal from common tail of adjacent sides $\vec{u}$ and $\vec{v}$. Form parallelogram; sum is the diagonal from shared starting point.
Diagonal from common tail of adjacent sides $\vec{u}$ and $\vec{v}$. Form parallelogram; sum is the diagonal from shared starting point.
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Find $\lVert\langle 6,8\rangle\rVert$.
Find $\lVert\langle 6,8\rangle\rVert$.
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$10$. Use $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
$10$. Use $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
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If $\vec{u}=\langle 2,1\rangle$ and $\vec{v}=\langle -1,5\rangle$, find $\vec{u}+\vec{v}$.
If $\vec{u}=\langle 2,1\rangle$ and $\vec{v}=\langle -1,5\rangle$, find $\vec{u}+\vec{v}$.
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$\langle 1,,6\rangle$. Add components: $\langle 2+(-1), 1+5\rangle = \langle 1, 6\rangle$.
$\langle 1,,6\rangle$. Add components: $\langle 2+(-1), 1+5\rangle = \langle 1, 6\rangle$.
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If $\vec{u}=\langle 4,0\rangle$ and $\vec{v}=\langle 0,3\rangle$, find $\lVert\vec{u}+\vec{v}\rVert$.
If $\vec{u}=\langle 4,0\rangle$ and $\vec{v}=\langle 0,3\rangle$, find $\lVert\vec{u}+\vec{v}\rVert$.
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$5$. First add to get $\langle 4,3\rangle$, then $\sqrt{4^2+3^2}=\sqrt{25}=5$.
$5$. First add to get $\langle 4,3\rangle$, then $\sqrt{4^2+3^2}=\sqrt{25}=5$.
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Compute $\lVert\langle 4,0\rangle\rVert+\lVert\langle 0,3\rangle\rVert$.
Compute $\lVert\langle 4,0\rangle\rVert+\lVert\langle 0,3\rangle\rVert$.
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$7$. Sum individual magnitudes: $\sqrt{16}+\sqrt{9}=4+3=7$.
$7$. Sum individual magnitudes: $\sqrt{16}+\sqrt{9}=4+3=7$.
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What inequality relates $\lVert\vec{u}+\vec{v}\rVert$ to $\lVert\vec{u}\rVert$ and $\lVert\vec{v}\rVert$?
What inequality relates $\lVert\vec{u}+\vec{v}\rVert$ to $\lVert\vec{u}\rVert$ and $\lVert\vec{v}\rVert$?
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$\lVert\vec{u}+\vec{v}\rVert\le\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$. The triangle inequality: sum magnitude ≤ magnitude sum.
$\lVert\vec{u}+\vec{v}\rVert\le\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$. The triangle inequality: sum magnitude ≤ magnitude sum.
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When does equality hold in $\lVert\vec{u}+\vec{v}\rVert\le\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$?
When does equality hold in $\lVert\vec{u}+\vec{v}\rVert\le\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$?
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When $\vec{u}$ and $\vec{v}$ point in the same direction. Equality occurs when vectors are parallel with same orientation.
When $\vec{u}$ and $\vec{v}$ point in the same direction. Equality occurs when vectors are parallel with same orientation.
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If $\vec{v}=\langle a,b\rangle$, what is $\vec{v}+(-\vec{v})$?
If $\vec{v}=\langle a,b\rangle$, what is $\vec{v}+(-\vec{v})$?
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$\vec{0}$. A vector plus its negative always gives the zero vector.
$\vec{0}$. A vector plus its negative always gives the zero vector.
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What property states that $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$?
What property states that $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$?
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Associative property of vector addition. Grouping doesn't affect the result when adding multiple vectors.
Associative property of vector addition. Grouping doesn't affect the result when adding multiple vectors.
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What is the component-wise sum formula for $\langle a,b\rangle+\langle c,d\rangle$?
What is the component-wise sum formula for $\langle a,b\rangle+\langle c,d\rangle$?
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$\langle a+c,,b+d\rangle$. Add corresponding components: first with first, second with second.
$\langle a+c,,b+d\rangle$. Add corresponding components: first with first, second with second.
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What property states that $\vec{u}+\vec{v}$ equals $\vec{v}+\vec{u}$?
What property states that $\vec{u}+\vec{v}$ equals $\vec{v}+\vec{u}$?
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Commutative property of vector addition. Vector addition order doesn't matter, like regular addition.
Commutative property of vector addition. Vector addition order doesn't matter, like regular addition.
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What is the triangle (end-to-end) rule for adding $\vec{u}$ and $\vec{v}$?
What is the triangle (end-to-end) rule for adding $\vec{u}$ and $\vec{v}$?
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Place tail of $\vec{v}$ at head of $\vec{u}$; sum is tail-to-head. Connect vectors tip-to-tail; resultant goes from start to end.
Place tail of $\vec{v}$ at head of $\vec{u}$; sum is tail-to-head. Connect vectors tip-to-tail; resultant goes from start to end.
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What is the magnitude formula for $\vec{v}=\langle a,b\rangle$?
What is the magnitude formula for $\vec{v}=\langle a,b\rangle$?
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$\lVert\vec{v}\rVert=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to the components.
$\lVert\vec{v}\rVert=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to the components.
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What is the vector from point $A(x_1,y_1)$ to point $B(x_2,y_2)$ in component form?
What is the vector from point $A(x_1,y_1)$ to point $B(x_2,y_2)$ in component form?
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$\langle x_2-x_1,,y_2-y_1\rangle$. Subtract initial point coordinates from terminal point coordinates.
$\langle x_2-x_1,,y_2-y_1\rangle$. Subtract initial point coordinates from terminal point coordinates.
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Identify the correct statement about magnitudes: $\lVert\vec{u}+\vec{v}\rVert$ vs. $\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$.
Identify the correct statement about magnitudes: $\lVert\vec{u}+\vec{v}\rVert$ vs. $\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$.
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Typically $\lVert\vec{u}+\vec{v}\rVert\ne\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$. Triangle inequality shows sum magnitude usually less than magnitude sum.
Typically $\lVert\vec{u}+\vec{v}\rVert\ne\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$. Triangle inequality shows sum magnitude usually less than magnitude sum.
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What is the additive identity for vectors (the vector $\vec{0}$) in $\mathbb{R}^2$?
What is the additive identity for vectors (the vector $\vec{0}$) in $\mathbb{R}^2$?
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$\vec{0}=\langle 0,0\rangle$. The zero vector has both components equal to zero.
$\vec{0}=\langle 0,0\rangle$. The zero vector has both components equal to zero.
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What is the additive inverse of $\vec{v}=\langle a,b\rangle$?
What is the additive inverse of $\vec{v}=\langle a,b\rangle$?
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$-\vec{v}=\langle -a,-b\rangle$. Negate both components to get the opposite vector.
$-\vec{v}=\langle -a,-b\rangle$. Negate both components to get the opposite vector.
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Find $\langle 3,-2\rangle+\langle -5,4\rangle$.
Find $\langle 3,-2\rangle+\langle -5,4\rangle$.
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$\langle -2,,2\rangle$. Add components: $3+(-5)=-2$ and $-2+4=2$.
$\langle -2,,2\rangle$. Add components: $3+(-5)=-2$ and $-2+4=2$.
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Find the vector $\overrightarrow{AB}$ for $A(1,-3)$ and $B(6,2)$.
Find the vector $\overrightarrow{AB}$ for $A(1,-3)$ and $B(6,2)$.
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$\langle 5,,5\rangle$. Calculate $\langle 6-1, 2-(-3)\rangle = \langle 5, 5\rangle$.
$\langle 5,,5\rangle$. Calculate $\langle 6-1, 2-(-3)\rangle = \langle 5, 5\rangle$.
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What does it mean to add vectors end-to-end (head-to-tail) in geometry?
What does it mean to add vectors end-to-end (head-to-tail) in geometry?
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Place tail of $\vec{v}$ at head of $\vec{u}$; sum is tail-to-head. Connect vectors tip-to-tail; resultant goes from start to end.
Place tail of $\vec{v}$ at head of $\vec{u}$; sum is tail-to-head. Connect vectors tip-to-tail; resultant goes from start to end.
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What is the component-wise subtraction formula for $\langle a,b\rangle - \langle c,d\rangle$?
What is the component-wise subtraction formula for $\langle a,b\rangle - \langle c,d\rangle$?
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$\langle a-c,,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.
$\langle a-c,,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.
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What is the magnitude formula for a vector $\vec{v}=\langle a,b\rangle$?
What is the magnitude formula for a vector $\vec{v}=\langle a,b\rangle$?
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$|\vec{v}|=\sqrt{a^2+b^2}$. Use the Pythagorean theorem on the components.
$|\vec{v}|=\sqrt{a^2+b^2}$. Use the Pythagorean theorem on the components.
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What is the parallelogram rule for $\vec{u}+\vec{v}$ in geometric vector addition?
What is the parallelogram rule for $\vec{u}+\vec{v}$ in geometric vector addition?
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Draw both from same tail; sum is diagonal of the parallelogram. Vectors form adjacent sides; sum is the diagonal.
Draw both from same tail; sum is diagonal of the parallelogram. Vectors form adjacent sides; sum is the diagonal.
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What is the commutative property of vector addition written with vectors $\vec{u}$ and $\vec{v}$?
What is the commutative property of vector addition written with vectors $\vec{u}$ and $\vec{v}$?
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$\vec{u}+\vec{v}=\vec{v}+\vec{u}$. Order doesn't matter in vector addition.
$\vec{u}+\vec{v}=\vec{v}+\vec{u}$. Order doesn't matter in vector addition.
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What is the associative property of vector addition for $\vec{u}$, $\vec{v}$, and $\vec{w}$?
What is the associative property of vector addition for $\vec{u}$, $\vec{v}$, and $\vec{w}$?
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$(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$. Grouping doesn't affect the result.
$(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$. Grouping doesn't affect the result.
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What is the additive identity vector, written as a component vector in $\mathbb{R}^2$?
What is the additive identity vector, written as a component vector in $\mathbb{R}^2$?
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$\langle 0,0\rangle$. The zero vector leaves any vector unchanged when added.
$\langle 0,0\rangle$. The zero vector leaves any vector unchanged when added.
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What is the additive inverse of $\vec{v}=\langle a,b\rangle$?
What is the additive inverse of $\vec{v}=\langle a,b\rangle$?
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$-\vec{v}=\langle -a,,-b\rangle$. Negate each component to get the opposite vector.
$-\vec{v}=\langle -a,,-b\rangle$. Negate each component to get the opposite vector.
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Find $\langle 3,-2\rangle + \langle -5,4\rangle$ using component-wise addition.
Find $\langle 3,-2\rangle + \langle -5,4\rangle$ using component-wise addition.
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$\langle -2,,2\rangle$. $3+(-5)=-2$ and $-2+4=2$.
$\langle -2,,2\rangle$. $3+(-5)=-2$ and $-2+4=2$.
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Find $\langle -1,7\rangle - \langle 4,-3\rangle$ using component-wise subtraction.
Find $\langle -1,7\rangle - \langle 4,-3\rangle$ using component-wise subtraction.
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$\langle -5,,10\rangle$. $-1-4=-5$ and $7-(-3)=10$.
$\langle -5,,10\rangle$. $-1-4=-5$ and $7-(-3)=10$.
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Find the magnitude $|\langle 6,8\rangle|$.
Find the magnitude $|\langle 6,8\rangle|$.
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$10$. $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
$10$. $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
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