Precalculus Flashcards: Find Sum Of Two Vectors

What this deck covers

This deck focuses on Find Sum Of Two Vectors, 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 the direction angle of $\langle -1,-1\rangle$ in radians?

Answer: $\frac{5\pi}{4}$. In quadrant III, add $\pi$ to reference angle $\frac{\pi}{4}$.

Flashcard 2: Identify the correct quadrant when $x<0$ and $y>0$ for a vector $\langle x,y\rangle$.

Answer: Quadrant $\mathrm{II}$. Negative $x$ and positive $y$ places the vector in the upper left quadrant.

Flashcard 3: State the magnitude of $\vec{u}+\vec{v}$ when magnitudes are $a,b$ and angle between is $\phi$.

Answer: $\sqrt{a^2+b^2+2ab\cos\phi}$. Law of cosines for vector addition with angle $\phi$ between vectors.

Flashcard 4: What is the magnitude of $\langle 3,4\rangle$?

Answer: $5$. Apply $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Flashcard 5: Find the magnitude and direction of the sum: $10$ at $0^\circ$ plus $10$ at $90^\circ$.

Answer: $10\sqrt{2},\ 45^\circ$. Right-angle sum gives diagonal: $\sqrt{10^2 + 10^2} = 10\sqrt{2}$ at $45°$.

Flashcard 6: What is the magnitude and direction of $5$ at $0^\circ$ plus $5$ at $180^\circ$?

Answer: $0,\ \text{direction undefined}$. Zero vector has no magnitude or direction.

Flashcard 7: Find the sum in magnitude-direction form: $1$ at $0$ plus $1$ at $\frac{\pi}{2}$.

Answer: Magnitude $\sqrt{2}$, direction $\frac{\pi}{4}$. Perpendicular unit vectors form right triangle: $\sqrt{1^2 + 1^2} = \sqrt{2}$.

Flashcard 8: What are the magnitude and direction of $\langle -3,4\rangle$ (give $\theta$ to nearest degree)?

Answer: $5,\ 127^\circ$. Negative x puts angle in quadrant II: $180° - 53° = 127°$.

Flashcard 9: Find the magnitude and direction of the sum: $2$ at $0^\circ$ plus $2$ at $90^\circ$.

Answer: $2\sqrt{2},\ 45^\circ$. Diagonal vector has magnitude $\sqrt{2^2+2^2} = 2\sqrt{2}$ at $45°$.

Flashcard 10: State the magnitude formula for a vector with components $\langle x,y\rangle$.

Answer: $\sqrt{x^2+y^2}$. Apply the Pythagorean theorem to find the length of the vector.

Flashcard 11: Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=6$ at $0^\circ$ and $\mathbf{v}=2$ at $180^\circ$.

Answer: $\langle 4,\ 0\rangle$. Right vector minus left vector: $6 - 2 = 4$ in x-direction.

Flashcard 12: Find the sum in magnitude-direction form: $2$ at $0$ plus $2$ at $\frac{\pi}{2}$.

Answer: Magnitude $2\sqrt{2}$, direction $\frac{\pi}{4}$. Sum is $\langle 2,2\rangle$ with magnitude $\sqrt{2^2 + 2^2} = 2\sqrt{2}$.

Flashcard 13: What is the direction angle of $\langle 1,-1\rangle$ in radians?

Answer: $\frac{7\pi}{4}$. In quadrant IV, subtract reference angle $\frac{\pi}{4}$ from $2\pi$.

Flashcard 14: Find the sum in magnitude-direction form: $2$ at $0$ plus $2$ at $\pi$.

Answer: Magnitude $0$, direction undefined. Opposite vectors cancel: $\langle 2,0\rangle + \langle -2,0\rangle = \langle 0,0\rangle$.

Flashcard 15: Find the sum in component form: $5$ at $0^\circ$ plus $5$ at $180^\circ$.

Answer: $\langle 0,\ 0\rangle$. Equal and opposite vectors cancel: $\langle 5,0\rangle + \langle -5,0\rangle$.

Flashcard 16: Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=4$ at $90^\circ$ and $\mathbf{v}=1$ at $90^\circ$.

Answer: $\langle 0,\ 5\rangle$. Both vectors point up, so $4 + 1 = 5$ in the y-direction.

Flashcard 17: Find the magnitude of the sum: $r$ at $0$ plus $r$ at $\frac{\pi}{2}$.

Answer: $r\sqrt{2}$. Perpendicular vectors of equal length form isosceles right triangle.

Flashcard 18: Find the sum in magnitude-direction form: $5$ at $\frac{\pi}{2}$ plus $12$ at $0$.

Answer: Magnitude $13$, direction $\operatorname{atan^2}(5,12)$. Forms 5-12-13 right triangle: $\langle 12,0\rangle + \langle 0,5\rangle = \langle 12,5\rangle$.

Flashcard 19: Identify the standard interval for a direction angle measured from the positive $x$-axis.

Answer: $0\le\theta<2\pi$. Standard convention measures counterclockwise from positive $x$-axis.

Flashcard 20: What is the direction angle of $\langle 1,1\rangle$ in radians?

Answer: $\frac{\pi}{4}$. Equal components give $45°$ angle, which is $\frac{\pi}{4}$ radians.

Flashcard 21: Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=5$ at $0^\circ$ and $\mathbf{v}=3$ at $0^\circ$.

Answer: $\langle 8,\ 0\rangle$. Both vectors point right, so $5 + 3 = 8$ in the x-direction.

Flashcard 22: Identify the standard direction angle (in degrees) of a vector pointing straight down.

Answer: $270^\circ$. Downward direction is $270°$ measured counterclockwise from positive x-axis.

Flashcard 23: Find the sum in component form: $2$ at $0^\circ$ plus $2$ at $90^\circ$.

Answer: $\langle 2,\ 2\rangle$. Right vector $\langle 2,0\rangle$ plus up vector $\langle 0,2\rangle$.

Flashcard 24: Find the sum in magnitude-direction form: $2$ at $0$ plus $3$ at $0$.

Answer: Magnitude $5$, direction $0$. Same direction vectors add magnitudes: $2 + 3 = 5$.

Flashcard 25: What are the magnitude and direction of $\langle 4,3\rangle$ (give $\theta$ to nearest degree)?

Answer: $5,\ 37^\circ$. Use $|\mathbf{v}| = 5$ and $\theta = \arctan(3/4) \approx 37°$.

Flashcard 26: State the component form of a vector with magnitude $r$ and direction angle $\theta$.

Answer: $\langle r\cos\theta,\ r\sin\theta\rangle$. Converts polar to rectangular using $x = r\cos\theta$ and $y = r\sin\theta$.

Flashcard 27: What are the magnitude and direction of $\langle 3,4\rangle$ (give $\theta$ to nearest degree)?

Answer: $5,\ 53^\circ$. Use $|\mathbf{v}| = 5$ and $\theta = \arctan(4/3) \approx 53°$.

Flashcard 28: Find $\mathbf{u}+\mathbf{v}$ in component form if $\mathbf{u}=3$ at $0^\circ$ and $\mathbf{v}=4$ at $90^\circ$.

Answer: $\langle 3,\ 4\rangle$. One vector points right (3), the other up (4).

Flashcard 29: What are the magnitude and direction of $\langle -4,-3\rangle$ (give $\theta$ to nearest degree)?

Answer: $5,\ 217^\circ$. Both components negative puts angle in quadrant III: $180° + 37° = 217°$.

Flashcard 30: State the magnitude of a vector $\langle x, y\rangle$ in the plane.

Answer: $\sqrt{x^2+y^2}$. Uses the Pythagorean theorem to find distance from origin.

Flashcard 31: What is the sum in component form of $\langle a,b\rangle+\langle c,d\rangle$?

Answer: $\langle a+c,\ b+d\rangle$. Add corresponding components: first with first, second with second.

Flashcard 32: Find the sum in magnitude-direction form: $3$ at $0$ plus $4$ at $\frac{\pi}{2}$.

Answer: Magnitude $5$, direction $\operatorname{atan^2}(4,3)$. Forms 3-4-5 right triangle: $\langle 3,0\rangle + \langle 0,4\rangle = \langle 3,4\rangle$.

Flashcard 33: What is the sum of vectors $\langle a,b\rangle$ and $\langle c,d\rangle$ in component form?

Answer: $\langle a+c,\ b+d\rangle$. Add corresponding components: first with first, second with second.

Flashcard 34: What is the direction angle of $\langle -1,1\rangle$ in radians?

Answer: $\frac{3\pi}{4}$. In quadrant II, add $\pi$ to reference angle $\frac{\pi}{4}$.