Find Sum of Two Vectors - Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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