Find Sum of Two Vectors - Pre-Calculus
Card 1 of 30
State the magnitude of a vector $\langle x, y\rangle$ in the plane.
State the magnitude of a vector $\langle x, y\rangle$ in the plane.
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$\sqrt{x^2+y^2}$. Uses the Pythagorean theorem to find distance from origin.
$\sqrt{x^2+y^2}$. Uses the Pythagorean theorem to find distance from origin.
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What is the sum of vectors $\langle a,b\rangle$ and $\langle c,d\rangle$ in component form?
What is the sum of vectors $\langle a,b\rangle$ and $\langle c,d\rangle$ in component form?
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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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Identify the standard interval for a direction angle measured from the positive $x$-axis.
Identify the standard interval for a direction angle measured from the positive $x$-axis.
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$0\le\theta<2\pi$. Standard convention measures counterclockwise from positive $x$-axis.
$0\le\theta<2\pi$. Standard convention measures counterclockwise from positive $x$-axis.
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Identify the correct quadrant when $x<0$ and $y>0$ for a vector $\langle x,y\rangle$.
Identify the correct quadrant when $x<0$ and $y>0$ for a vector $\langle x,y\rangle$.
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Quadrant $\mathrm{II}$. Negative $x$ and positive $y$ places the vector in the upper left quadrant.
Quadrant $\mathrm{II}$. Negative $x$ and positive $y$ places the vector in the upper left quadrant.
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What is the magnitude of $\langle 3,4\rangle$?
What is the magnitude of $\langle 3,4\rangle$?
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$5$. Apply $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
$5$. Apply $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
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Find the sum in magnitude-direction form: $2$ at $0$ plus $2$ at $\frac{\pi}{2}$.
Find the sum in magnitude-direction form: $2$ at $0$ plus $2$ at $\frac{\pi}{2}$.
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Magnitude $2\sqrt{2}$, direction $\frac{\pi}{4}$. Sum is $\langle 2,2\rangle$ with magnitude $\sqrt{2^2 + 2^2} = 2\sqrt{2}$.
Magnitude $2\sqrt{2}$, direction $\frac{\pi}{4}$. Sum is $\langle 2,2\rangle$ with magnitude $\sqrt{2^2 + 2^2} = 2\sqrt{2}$.
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Find the sum in magnitude-direction form: $1$ at $0$ plus $1$ at $\frac{\pi}{2}$.
Find the sum in magnitude-direction form: $1$ at $0$ plus $1$ at $\frac{\pi}{2}$.
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Magnitude $\sqrt{2}$, direction $\frac{\pi}{4}$. Perpendicular unit vectors form right triangle: $\sqrt{1^2 + 1^2} = \sqrt{2}$.
Magnitude $\sqrt{2}$, direction $\frac{\pi}{4}$. Perpendicular unit vectors form right triangle: $\sqrt{1^2 + 1^2} = \sqrt{2}$.
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Find the sum in magnitude-direction form: $2$ at $0$ plus $2$ at $\pi$.
Find the sum in magnitude-direction form: $2$ at $0$ plus $2$ at $\pi$.
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Magnitude $0$, direction undefined. Opposite vectors cancel: $\langle 2,0\rangle + \langle -2,0\rangle = \langle 0,0\rangle$.
Magnitude $0$, direction undefined. Opposite vectors cancel: $\langle 2,0\rangle + \langle -2,0\rangle = \langle 0,0\rangle$.
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Find the sum in magnitude-direction form: $2$ at $0$ plus $3$ at $0$.
Find the sum in magnitude-direction form: $2$ at $0$ plus $3$ at $0$.
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Magnitude $5$, direction $0$. Same direction vectors add magnitudes: $2 + 3 = 5$.
Magnitude $5$, direction $0$. Same direction vectors add magnitudes: $2 + 3 = 5$.
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What is the direction angle of $\langle 1,-1\rangle$ in radians?
What is the direction angle of $\langle 1,-1\rangle$ in radians?
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$\frac{7\pi}{4}$. In quadrant IV, subtract reference angle $\frac{\pi}{4}$ from $2\pi$.
$\frac{7\pi}{4}$. In quadrant IV, subtract reference angle $\frac{\pi}{4}$ from $2\pi$.
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What is the direction angle of $\langle -1,-1\rangle$ in radians?
What is the direction angle of $\langle -1,-1\rangle$ in radians?
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$\frac{5\pi}{4}$. In quadrant III, add $\pi$ to reference angle $\frac{\pi}{4}$.
$\frac{5\pi}{4}$. In quadrant III, add $\pi$ to reference angle $\frac{\pi}{4}$.
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What is the direction angle of $\langle -1,1\rangle$ in radians?
What is the direction angle of $\langle -1,1\rangle$ in radians?
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$\frac{3\pi}{4}$. In quadrant II, add $\pi$ to reference angle $\frac{\pi}{4}$.
$\frac{3\pi}{4}$. In quadrant II, add $\pi$ to reference angle $\frac{\pi}{4}$.
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What is the direction angle of $\langle 1,1\rangle$ in radians?
What is the direction angle of $\langle 1,1\rangle$ in radians?
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$\frac{\pi}{4}$. Equal components give $45°$ angle, which is $\frac{\pi}{4}$ radians.
$\frac{\pi}{4}$. Equal components give $45°$ angle, which is $\frac{\pi}{4}$ radians.
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State the magnitude of $\vec{u}+\vec{v}$ when magnitudes are $a,b$ and angle between is $\phi$.
State the magnitude of $\vec{u}+\vec{v}$ when magnitudes are $a,b$ and angle between is $\phi$.
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$\sqrt{a^2+b^2+2ab\cos\phi}$. Law of cosines for vector addition with angle $\phi$ between vectors.
$\sqrt{a^2+b^2+2ab\cos\phi}$. Law of cosines for vector addition with angle $\phi$ between vectors.
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Find the magnitude of the sum: $r$ at $0$ plus $r$ at $\frac{\pi}{2}$.
Find the magnitude of the sum: $r$ at $0$ plus $r$ at $\frac{\pi}{2}$.
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$r\sqrt{2}$. Perpendicular vectors of equal length form isosceles right triangle.
$r\sqrt{2}$. Perpendicular vectors of equal length form isosceles right triangle.
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Find the sum in magnitude-direction form: $5$ at $\frac{\pi}{2}$ plus $12$ at $0$.
Find the sum in magnitude-direction form: $5$ at $\frac{\pi}{2}$ plus $12$ at $0$.
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Magnitude $13$, direction $\operatorname{atan^2}(5,12)$. Forms 5-12-13 right triangle: $\langle 12,0\rangle + \langle 0,5\rangle = \langle 12,5\rangle$.
Magnitude $13$, direction $\operatorname{atan^2}(5,12)$. Forms 5-12-13 right triangle: $\langle 12,0\rangle + \langle 0,5\rangle = \langle 12,5\rangle$.
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Find the sum in magnitude-direction form: $3$ at $0$ plus $4$ at $\frac{\pi}{2}$.
Find the sum in magnitude-direction form: $3$ at $0$ plus $4$ at $\frac{\pi}{2}$.
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Magnitude $5$, direction $\operatorname{atan^2}(4,3)$. Forms 3-4-5 right triangle: $\langle 3,0\rangle + \langle 0,4\rangle = \langle 3,4\rangle$.
Magnitude $5$, direction $\operatorname{atan^2}(4,3)$. Forms 3-4-5 right triangle: $\langle 3,0\rangle + \langle 0,4\rangle = \langle 3,4\rangle$.
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State the component form of a vector with magnitude $r$ and direction angle $\theta$.
State the component form of a vector with magnitude $r$ and direction angle $\theta$.
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$\langle r\cos\theta,\ r\sin\theta\rangle$. Converts polar to rectangular using $x = r\cos\theta$ and $y = r\sin\theta$.
$\langle r\cos\theta,\ r\sin\theta\rangle$. Converts polar to rectangular using $x = r\cos\theta$ and $y = r\sin\theta$.
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Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=5$ at $0^\circ$ and $\mathbf{v}=3$ at $0^\circ$.
Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=5$ at $0^\circ$ and $\mathbf{v}=3$ at $0^\circ$.
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$\langle 8,\ 0\rangle$. Both vectors point right, so $5 + 3 = 8$ in the x-direction.
$\langle 8,\ 0\rangle$. Both vectors point right, so $5 + 3 = 8$ in the x-direction.
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State the magnitude formula for a vector with components $\langle x,y\rangle$.
State the magnitude formula for a vector with components $\langle x,y\rangle$.
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$\sqrt{x^2+y^2}$. Apply the Pythagorean theorem to find the length of the vector.
$\sqrt{x^2+y^2}$. Apply the Pythagorean theorem to find the length of the vector.
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What is the sum in component form of $\langle a,b\rangle+\langle c,d\rangle$?
What is the sum in component form of $\langle a,b\rangle+\langle c,d\rangle$?
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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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Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=4$ at $90^\circ$ and $\mathbf{v}=1$ at $90^\circ$.
Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=4$ at $90^\circ$ and $\mathbf{v}=1$ at $90^\circ$.
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$\langle 0,\ 5\rangle$. Both vectors point up, so $4 + 1 = 5$ in the y-direction.
$\langle 0,\ 5\rangle$. Both vectors point up, so $4 + 1 = 5$ in the y-direction.
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Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=6$ at $0^\circ$ and $\mathbf{v}=2$ at $180^\circ$.
Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=6$ at $0^\circ$ and $\mathbf{v}=2$ at $180^\circ$.
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$\langle 4,\ 0\rangle$. Right vector minus left vector: $6 - 2 = 4$ in x-direction.
$\langle 4,\ 0\rangle$. Right vector minus left vector: $6 - 2 = 4$ in x-direction.
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Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=3$ at $0^\circ$ and $\mathbf{v}=4$ at $90^\circ$.
Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=3$ at $0^\circ$ and $\mathbf{v}=4$ at $90^\circ$.
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$\langle 3,\ 4\rangle$. One vector points right (3), the other up (4).
$\langle 3,\ 4\rangle$. One vector points right (3), the other up (4).
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What are the magnitude and direction of $\langle 3,4\rangle$ (give $\theta$ to nearest degree)?
What are the magnitude and direction of $\langle 3,4\rangle$ (give $\theta$ to nearest degree)?
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$5,\ 53^\circ$. Use $|\mathbf{v}| = 5$ and $\theta = \arctan(4/3) \approx 53°$.
$5,\ 53^\circ$. Use $|\mathbf{v}| = 5$ and $\theta = \arctan(4/3) \approx 53°$.
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What are the magnitude and direction of $\langle 4,3\rangle$ (give $\theta$ to nearest degree)?
What are the magnitude and direction of $\langle 4,3\rangle$ (give $\theta$ to nearest degree)?
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$5,\ 37^\circ$. Use $|\mathbf{v}| = 5$ and $\theta = \arctan(3/4) \approx 37°$.
$5,\ 37^\circ$. Use $|\mathbf{v}| = 5$ and $\theta = \arctan(3/4) \approx 37°$.
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What are the magnitude and direction of $\langle -3,4\rangle$ (give $\theta$ to nearest degree)?
What are the magnitude and direction of $\langle -3,4\rangle$ (give $\theta$ to nearest degree)?
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$5,\ 127^\circ$. Negative x puts angle in quadrant II: $180° - 53° = 127°$.
$5,\ 127^\circ$. Negative x puts angle in quadrant II: $180° - 53° = 127°$.
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What are the magnitude and direction of $\langle -4,-3\rangle$ (give $\theta$ to nearest degree)?
What are the magnitude and direction of $\langle -4,-3\rangle$ (give $\theta$ to nearest degree)?
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$5,\ 217^\circ$. Both components negative puts angle in quadrant III: $180° + 37° = 217°$.
$5,\ 217^\circ$. Both components negative puts angle in quadrant III: $180° + 37° = 217°$.
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Find the sum in component form: $2$ at $0^\circ$ plus $2$ at $90^\circ$.
Find the sum in component form: $2$ at $0^\circ$ plus $2$ at $90^\circ$.
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$\langle 2,\ 2\rangle$. Right vector $\langle 2,0\rangle$ plus up vector $\langle 0,2\rangle$.
$\langle 2,\ 2\rangle$. Right vector $\langle 2,0\rangle$ plus up vector $\langle 0,2\rangle$.
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Find the magnitude and direction of the sum: $2$ at $0^\circ$ plus $2$ at $90^\circ$.
Find the magnitude and direction of the sum: $2$ at $0^\circ$ plus $2$ at $90^\circ$.
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$2\sqrt{2},\ 45^\circ$. Diagonal vector has magnitude $\sqrt{2^2+2^2} = 2\sqrt{2}$ at $45°$.
$2\sqrt{2},\ 45^\circ$. Diagonal vector has magnitude $\sqrt{2^2+2^2} = 2\sqrt{2}$ at $45°$.
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