Finding Components of Vectors - Precalculus

1. State the formula for the component form of the vector from $P(x_1,y_1)$ to $Q(x_2,y_2)$.

   $\langle x_2-x_1,\,y_2-y_1\rangle$. Terminal minus initial gives the displacement vector.

2. What is the component form of the vector from $P(-10,4)$ to $Q(-3,12)$?

   $\langle 7,\ 8\rangle$. Calculate $\langle -3-(-10), 12-4\rangle = \langle 7, 8\rangle$.

3. What operation gives vector components from initial point $P$ to terminal point $Q$ in the plane?

   Subtract coordinates: $Q-P$. Terminal coordinates minus initial coordinates gives components.

4. What is the component form of the vector from $P(-5,4)$ to $Q(-9,10)$?

   $\langle -4,\,6\rangle$. Calculate: $\langle -9-(-5), 10-4\rangle = \langle -4, 6\rangle$.

5. What is the component form of the vector from $P(2,-1)$ to $Q(7,3)$?

   $\langle 5,\,4\rangle$. Calculate: $\langle 7-2, 3-(-1)\rangle = \langle 5, 4\rangle$.

6. What is the component form of the vector from $P(0,0)$ to $Q(-3,5)$?

   $\langle -3,\,5\rangle$. Calculate: $\langle -3-0, 5-0\rangle = \langle -3, 5\rangle$.

7. What operation produces vector components from initial $P(x_1,y_1)$ to terminal $Q(x_2,y_2)$?

   Subtract initial coordinates from terminal: $Q-P$. Vector components are found by terminal minus initial coordinates.

8. If $\overrightarrow{PQ}=\langle a,b\rangle$, what is $\overrightarrow{QP}$ in component form?

   $\langle -a,\ -b\rangle$. Reversing direction negates both components.

9. Which vector is the opposite direction of $\overrightarrow{PQ}$: $\overrightarrow{QP}$ or $\overrightarrow{PQ}$?

   $\overrightarrow{QP}$. Reversing initial and terminal points reverses the vector.

10. Which option is the correct component form for $\overrightarrow{AB}$ if $A(-1,2)$ and $B(3,-5)$?

    $\langle 4,\,-7\rangle$. Calculate: $\langle 3-(-1), -5-2\rangle = \langle 4, -7\rangle$.

11. Find and correct the error: $A(1,4)$, $B(6,0)$, claimed $\overrightarrow{AB}=\langle -5,4\rangle$.

    Correct: $\overrightarrow{AB}=\langle 5,\,-4\rangle$. Should be $\langle 6-1, 0-4\rangle = \langle 5, -4\rangle$, not negated.

12. What is the component form of the vector from $P(5,5)$ to $Q(5,-2)$?

    $\langle 0,\ -7\rangle$. Calculate $\langle 5-5, -2-5\rangle = \langle 0, -7\rangle$.

13. What is $\overrightarrow{QP}$ in component form if $\overrightarrow{PQ}=\langle 7,-2\rangle$?

    $\langle -7,\,2\rangle$. Reverse vector has opposite components: negate each component.

14. What is the component form of the vector from $P\left(\frac{1}{2},-3\right)$ to $Q\left(\frac{5}{2},1\right)$?

    $\langle 2,\ 4\rangle$. Calculate $\langle \frac{5}{2}-\frac{1}{2}, 1-(-3)\rangle = \langle 2, 4\rangle$.

15. What is the component form of the vector from $P(-6,0)$ to $Q(-1,-5)$?

    $\langle 5,\ -5\rangle$. Calculate $\langle -1-(-6), -5-0\rangle = \langle 5, -5\rangle$.

16. Identify the error: a student wrote $\overrightarrow{PQ}=\langle x_1-x_2,\ y_1-y_2\rangle$ for $P\to Q$.

    Correct is $\langle x_2-x_1,\ y_2-y_1\rangle$. Student reversed the subtraction order.

17. Find the initial point $P$ if $Q(5,-4)$ and $\overrightarrow{PQ}=\langle 2,3\rangle$.

    $P(3,-7)$. Subtract vector from terminal: $(5,-4) - \langle 2,3\rangle = (3,-7)$.

18. Find $Q$ if $P(1,-2)$ and $\overrightarrow{PQ}=\langle 4,5\rangle$.

    $Q(5,3)$. Add vector components to initial point: $Q = (1+4, -2+5)$.

19. What is the component form of the vector from $P(a,b)$ to $Q(c,d)$?

    $\langle c-a,\,d-b\rangle$. General formula: terminal minus initial coordinates.

20. Identify the component form of the vector from $P(-3,1)$ to $Q(2,1)$.

    $\langle 5,\,0\rangle$. Calculate: $\langle 2-(-3), 1-1\rangle = \langle 5, 0\rangle$.

21. Identify the component form of the vector from $P(6,-3)$ to $Q(6,-9)$.

    $\langle 0,\,-6\rangle$. Calculate: $\langle 6-6, -9-(-3)\rangle = \langle 0, -6\rangle$.

22. What is the component form of the vector from $P(-2,-3)$ to $Q(-2,4)$?

    $\langle 0,\ 7\rangle$. Calculate $\langle -2-(-2), 4-(-3)\rangle = \langle 0, 7\rangle$.

23. What is the component form of the vector from $P(9,-1)$ to $Q(2,-1)$?

    $\langle -7,\ 0\rangle$. Calculate $\langle 2-9, -1-(-1)\rangle = \langle -7, 0\rangle$.

24. What is the component form of the vector from $P(-2,-7)$ to $Q(-2,1)$?

    $\langle 0,\,8\rangle$. Calculate: $\langle -2-(-2), 1-(-7)\rangle = \langle 0, 8\rangle$.

25. What is the component form of the vector from $P(0,0)$ to $Q(-3,8)$?

    $\langle -3,\ 8\rangle$. Calculate $\langle -3-0, 8-0\rangle = \langle -3, 8\rangle$.

26. Find $P$ if $Q(-1,6)$ and $\overrightarrow{PQ}=\langle 3,-2\rangle$.

    $P(-4,8)$. Subtract vector from terminal: $P = (-1-3, 6-(-2))$.

27. Find the terminal point $Q$ if $P(2,3)$ and $\overrightarrow{PQ}=\langle 4,-1\rangle$.

    $Q(6,2)$. Add vector to initial point: $(2,3) + \langle 4,-1\rangle = (6,2)$.

28. What is the component form of the vector from $P(3,7)$ to $Q(-4,1)$?

    $\langle -7,\ -6\rangle$. Calculate $\langle -4-3, 1-7\rangle = \langle -7, -6\rangle$.

29. What is the component form of the vector from $P(1.5,-2)$ to $Q(4.5,1)$?

    $\langle 3,\,3\rangle$. Calculate: $\langle 4.5-1.5, 1-(-2)\rangle = \langle 3, 3\rangle$.

30. What is the component form of the vector from $P(-4,6)$ to $Q(1,2)$?

    $\langle 5,\ -4\rangle$. Calculate $\langle 1-(-4), 2-6\rangle = \langle 5, -4\rangle$.

31. Identify the $x$-component of the vector from $P(x_1,y_1)$ to $Q(x_2,y_2)$.

    $x_2-x_1$. The horizontal displacement from initial to terminal point.

32. Identify the $y$-component of the vector from $P(x_1,y_1)$ to $Q(x_2,y_2)$.

    $y_2-y_1$. The vertical displacement from initial to terminal point.

33. What is the component form of the vector from $P(9,2)$ to $Q(4,2)$?

    $\langle -5,\,0\rangle$. Calculate: $\langle 4-9, 2-2\rangle = \langle -5, 0\rangle$.

34. What is the component form of the vector from $P(-4,6)$ to $Q(1,-2)$?

    $\langle 5,\,-8\rangle$. Calculate: $\langle 1-(-4), -2-6\rangle = \langle 5, -8\rangle$.

35. What is the component form of the vector from $P(-1.5,2)$ to $Q(0.5,-1)$?

    $\langle 2,\ -3\rangle$. Calculate $\langle 0.5-(-1.5), -1-2\rangle = \langle 2, -3\rangle$.

36. What is the component form of the vector from $P(3,8)$ to $Q(-1,0)$?

    $\langle -4,\,-8\rangle$. Calculate: $\langle -1-3, 0-8\rangle = \langle -4, -8\rangle$.

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