Question

drag the tiles to the correct boxes to complete the pairs.
given: ( mathbf{u} = langle -2, -7
angle ), ( mathbf{v} = langle 3, -4
angle ), and ( mathbf{w} = langle 5, 1
angle ).
based on the components of the given vectors, match each vector subtraction with the graph of the resulting vector.
graphs (graph 1, graph 2, graph 3, graph 4) showing vectors on coordinate planes are present but omitted from ocr text as per instructions

Explanation:

Step1: Recall Vector Subtraction

Vector subtraction \(\mathbf{a} - \mathbf{b}=\langle a_1 - b_1,a_2 - b_2
angle\) for \(\mathbf{a}=\langle a_1,a_2
angle,\mathbf{b}=\langle b_1,b_2
angle\).

Step2: Calculate \( \mathbf{u}-\mathbf{v} \)

\(\mathbf{u}=\langle - 2,-7
angle,\mathbf{v}=\langle3,-4
angle\)
\(\mathbf{u}-\mathbf{v}=\langle - 2 - 3,-7-(-4)
angle=\langle - 5,-3
angle\)
Check graphs: Graph 1 has a vector from origin to \((-5,-3)\)? Wait, Graph 1's vector: from origin? Wait, no, the first two graphs have vectors starting at some point? Wait, no, maybe the graphs are of the resulting vector. Wait, Graph 1: the red vector goes from (0,0) to (-5,-3)? Wait, no, looking at Graph 1: the x - axis, the vector is from (0,0) to (-5,-3)? Wait, no, the first graph (Graph 1) has a vector with endpoint at (-5,-3)? Wait, no, the first graph: x from -6 to 4, y from -8 to 6. The red vector is from (0,0) to (-5,-3)? Wait, no, maybe I misread. Wait, let's recalculate other subtractions.

Step3: Calculate \( \mathbf{v}-\mathbf{u} \)

\(\mathbf{v}-\mathbf{u}=\langle3-(-2),-4 - (-7)
angle=\langle5,3
angle\)
Graph 3: vector from (0,0) to (5,3)? Wait, Graph 3 has a vector to (2,5)? No, wait Graph 3: x from -4 to 8, y from -4 to 6. The red vector is to (2,5)? Wait, no, maybe \( \mathbf{w}-\mathbf{v} \):

Step4: Calculate \( \mathbf{w}-\mathbf{v} \)

\(\mathbf{w}=\langle5,1
angle,\mathbf{v}=\langle3,-4
angle\)
\(\mathbf{w}-\mathbf{v}=\langle5 - 3,1-(-4)
angle=\langle2,5
angle\)
Graph 3: vector to (2,5)? Yes, Graph 3's vector is to (2,5).

Step5: Calculate \( \mathbf{w}-\mathbf{u} \)

\(\mathbf{w}-\mathbf{u}=\langle5-(-2),1-(-7)
angle=\langle7,8
angle\)? No, wait \(\mathbf{w}=\langle5,1
angle,\mathbf{u}=\langle - 2,-7
angle\)
\(\mathbf{w}-\mathbf{u}=\langle5 + 2,1 + 7
angle=\langle7,8
angle\)? No, Graph 4: vector to (7,8)? No, Graph 4's vector is more shallow. Wait, maybe \( \mathbf{v}-\mathbf{w} \):

\(\mathbf{v}-\mathbf{w}=\langle3 - 5,-4 - 1
angle=\langle - 2,-5
angle\)? No, let's re - check.

Wait, maybe the problem is to match each subtraction (like \(\mathbf{u}-\mathbf{v},\mathbf{v}-\mathbf{u},\mathbf{w}-\mathbf{v},\mathbf{w}-\mathbf{u}\)) to the graphs. Let's take \(\mathbf{u}-\mathbf{v}=\langle - 5,-3
angle\): which graph has a vector with components (-5,-3)? Graph 1: the vector in Graph 1: from (0,0) to (-5,-3)? Wait, Graph 1's vector: x - component - 5, y - component - 3. Graph 2: vector with x - component - 6, y - component - 8? No. \(\mathbf{v}-\mathbf{u}=\langle5,3
angle\): Graph 4? No, Graph 3: vector to (2,5)? Wait, \(\mathbf{w}-\mathbf{v}=\langle2,5
angle\) (since \(5 - 3 = 2,1-(-4)=5\)): Graph 3 has a vector with x = 2, y = 5. \(\mathbf{w}-\mathbf{u}=\langle5-(-2),1 - (-7)
angle=\langle7,8
angle\)? No, \(\mathbf{u}-\mathbf{w}=\langle - 2-5,-7 - 1
angle=\langle - 7,-8
angle\): Graph 2 has a vector with x=-7,y=-8? Yes, Graph 2's vector: from (0,0) to (-7,-8)?

Let's formalize:

1. \(\mathbf{u}-\mathbf{v}=\langle - 2-3,-7 - (-4)

angle=\langle - 5,-3
angle\) → Graph 1 (vector with x=-5,y=-3)

1. \(\mathbf{u}-\mathbf{w}=\langle - 2 - 5,-7 - 1

angle=\langle - 7,-8
angle\) → Graph 2 (vector with x=-7,y=-8)

1. \(\mathbf{w}-\mathbf{v}=\langle5 - 3,1-(-4)

angle=\langle2,5
angle\) → Graph 3 (vector with x=2,y=5)

1. \(\mathbf{v}-\mathbf{u}=\langle3-(-2),-4 - (-7)

angle=\langle5,3
angle\) → Graph 4 (vector with x=5,y=3)

Answer:

For example, if the subtractions are \(\mathbf{u}-\mathbf{v}\) → Graph 1, \(\mathbf{u}-\mathbf{w}\) → Graph 2, \(\mathbf{w}-\mathbf{v}\) → Graph 3, \(\mathbf{v}-\mathbf{u}\) → Graph 4 (depending on the exact graph labels, but following the vector component calculations).