Subtract and Represent Vectors Graphically - Precalculus

1. What is $\mathbf{v}-\mathbf{0}$ for any vector $\mathbf{v}$?

   $\mathbf{v}$. Subtracting the zero vector leaves any vector unchanged.

2. If $\mathbf{v}=\langle 2,1\rangle$ and $\mathbf{w}=\langle -4,3\rangle$, what is $\mathbf{v}-\mathbf{w}$?

   $\langle 6,-2\rangle$. Subtract components: $2-(-4)=6$ and $1-3=-2$.

3. What is $\overrightarrow{PQ}$ if $P(-3,4)$ and $Q(2,9)$?

   $\overrightarrow{PQ}=\langle 5,5\rangle$. Vector from $P$ to $Q$ is $Q-P$: $(2-(-3), 9-4)$.

4. In a tip-to-tail diagram for $\mathbf{v}+(-\mathbf{w})$, what does the resultant connect?

   From tail of $\mathbf{v}$ to tip of $-\mathbf{w}$. The resultant spans from start to end of the path.

5. If $\mathbf{v}=\langle -5,0\rangle$ and $\mathbf{w}=\langle 1,-7\rangle$, what is $\mathbf{v}-\mathbf{w}$?

   $\langle -6,7\rangle$. Subtract components: $-5-1=-6$ and $0-(-7)=7$.

6. Which vector equals $\mathbf{w}-\mathbf{v}$ in terms of $\mathbf{v}-\mathbf{w}$?

   $\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.

7. State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2,v_3\rangle$ and $\mathbf{w}=\langle w_1,w_2,w_3\rangle$.

   $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,\,v_2-w_2,\,v_3-w_3\rangle$. Subtract each corresponding component in 3D.

8. What is the identity that rewrites vector subtraction $\mathbf{v}-\mathbf{w}$ using addition?

   $\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction becomes addition of the additive inverse.

9. What is $\langle -1,4\rangle-\langle 6,-3\rangle$?

   $\langle -7,7\rangle$. Apply component-wise subtraction: $-1-6=-7$ and $4-(-3)=7$.

10. What is $\mathbf{0}-\mathbf{w}$ for any vector $\mathbf{w}$?

    $-\mathbf{w}$. Zero minus any vector equals its negative.

11. Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle -3,8\rangle$ and $\mathbf{w}=\langle 1,-6\rangle$.

    $\langle -4,14\rangle$. $\langle -3-1, 8-(-6)\rangle = \langle -4, 8+6\rangle$.

12. Identify the correct simplification of $\mathbf{v}-(-\mathbf{w})$.

    $\mathbf{v}+\mathbf{w}$. Double negative becomes positive: $-(-\mathbf{w}) = \mathbf{w}$.

13. Find $\mathbf{v}$ if $\mathbf{v}-\langle 3,-2\rangle=\langle -1,5\rangle$.

    $\mathbf{v}=\langle 2,3\rangle$. Add $\langle 3,-2\rangle$ to both sides: $\mathbf{v}=\langle -1,5\rangle+\langle 3,-2\rangle$.

14. What is $\mathbf{v}-\mathbf{v}$ for any vector $\mathbf{v}$?

    $\mathbf{0}$. Any vector minus itself equals the zero vector.

15. What is the definition of vector subtraction $\mathbf{v}-\mathbf{w}$ in terms of addition?

    $\mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})$. Subtraction is defined as adding the additive inverse.

16. Which expression is equivalent to $\mathbf{w}-\mathbf{v}$ written using $\mathbf{v}-\mathbf{w}$?

    $\mathbf{w}-\mathbf{v}=-(\mathbf{v}-\mathbf{w})$. Reversing subtraction order negates the result.

17. Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle 4,1\rangle$ and $\mathbf{w}=\langle -2,5\rangle$.

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

18. What is the magnitude relationship between $\mathbf{w}$ and $-\mathbf{w}$?

    $\lVert -\mathbf{w}\rVert=\lVert \mathbf{w}\rVert$. Negating a vector preserves its magnitude.

19. What is $\mathbf{v}-\mathbf{0}$ for any vector $\mathbf{v}$?

    $\mathbf{v}$. Subtracting the zero vector leaves any vector unchanged.

20. When $\mathbf{v}$ and $\mathbf{w}$ share the same tail, what geometric vector equals $\mathbf{v}-\mathbf{w}$?

    The vector from tip of $\mathbf{w}$ to tip of $\mathbf{v}$. Direct path from $\mathbf{w}$'s endpoint to $\mathbf{v}$'s endpoint.

21. State the component-wise subtraction rule for $\mathbf{v}=\langle v_1,v_2\rangle$ and $\mathbf{w}=\langle w_1,w_2\rangle$.

    $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,\,v_2-w_2\rangle$. Subtract corresponding components.

22. Find and correct the error: $\langle a,b\rangle-\langle c,d\rangle=\langle a+c,\,b+d\rangle$.

    Correct: $\langle a-c,\,b-d\rangle$. The error shows addition; subtraction requires $a-c$ and $b-d$.

23. What is $-\langle -8,2\rangle$?

    $\langle 8,-2\rangle$. Negate each component: $-(-8)=8$ and $-(2)=-2$.

24. What is the component-wise subtraction rule for $\langle a,b\rangle-\langle c,d\rangle$?

    $\langle a,b\rangle-\langle c,d\rangle=\langle a-c,\,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.

25. Compute $\mathbf{v}-\mathbf{w}$ if $\mathbf{v}=\langle 2,-1,5\rangle$ and $\mathbf{w}=\langle -4,3,2\rangle$.

    $\langle 6,-4,3\rangle$. $\langle 2-(-4), -1-3, 5-2\rangle = \langle 6, -4, 3\rangle$.

26. Which vector should you add to $\mathbf{v}$ to compute $\mathbf{v}-\mathbf{w}$ graphically?

    $-\mathbf{w}$. Use the additive inverse to convert subtraction to addition.

27. What is $\langle 7,-2\rangle-\langle 3,5\rangle$?

    $\langle 4,-7\rangle$. Apply component-wise subtraction: $7-3=4$ and $-2-5=-7$.

28. Find $-\mathbf{w}$ if $\mathbf{w}=\langle 7,-3\rangle$.

    $\langle -7,3\rangle$. Negate each component: $7$ becomes $-7$, $-3$ becomes $3$.

29. What are the magnitude and direction of $-\mathbf{w}$ compared with $\mathbf{w}$?

    Same magnitude, opposite direction. Negating a vector reverses its direction but preserves its length.

30. What is the vector from point $A(1,2)$ to point $B(6,-1)$ written as $\overrightarrow{AB}$?

    $\overrightarrow{AB}=\langle 5,-3\rangle$. Vector from $A$ to $B$ is $B-A$: $(6-1, -1-2)$.

31. Find and correct the error: A student wrote $\mathbf{v}-\mathbf{w}=\langle v_1-w_2,\,v_2-w_1\rangle$.

    Correct: $\mathbf{v}-\mathbf{w}=\langle v_1-w_1,\,v_2-w_2\rangle$. Student mixed indices; subtract matching components.

32. What is the component form of $-\langle a,b\rangle$?

    $-\langle a,b\rangle=\langle -a,-b\rangle$. Negate each component to find the additive inverse.

33. What is the direction relationship between $\mathbf{w}$ and $-\mathbf{w}$?

    $-\mathbf{w}$ points opposite $\mathbf{w}$. Negating reverses the vector's direction by 180°.

34. What is the additive inverse of a vector $\mathbf{w}$ called and written as?

    Additive inverse: $-\mathbf{w}$. The negative of a vector is its additive inverse.

35. What is $\mathbf{v}-\mathbf{v}$ for any vector $\mathbf{v}$?

    $\mathbf{0}$. Any vector minus itself equals the zero vector.

36. What is the additive inverse of a vector $\mathbf{w}$ written in component form?

    If $\mathbf{w}=\langle a,b\rangle$, then $-\mathbf{w}=\langle -a,-b\rangle$. Negate each component to get the additive inverse.

37. What is $\mathbf{0}-\mathbf{w}$ for any vector $\mathbf{w}$?

    $-\mathbf{w}$. Zero minus a vector equals the negative of that vector.

38. Identify the vector represented by the arrow from the tip of $\mathbf{w}$ to the tip of $\mathbf{v}$ (tails together).

    $\mathbf{v}-\mathbf{w}$. The difference vector points from $\mathbf{w}$'s tip to $\mathbf{v}$'s tip.

39. In a tip-to-tail diagram, how do you place $-\mathbf{w}$ to represent $\mathbf{v}-\mathbf{w}$?

    Place tail of $-\mathbf{w}$ at tip of $\mathbf{v}$. Standard tip-to-tail addition method.

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