Question

drag each tile to the correct box. arrange the vector operations in increasing order of the magnitude of the resulting vectors. tiles a + (-b), where a = <-3, 5> and b = <2, -1> c + (-d), where c = <2, 7> and d = <3, 5> e + (-f), where e = <4, 1> and f = <7, 4> g + (-h), where g = <3, -3> and h = <5, -1> m + (-n), where m = <2, -5> and n = <-2, 4> o + (-p), where o = <-1, -5> and p = <3, -8> sequence boxes for sequence with arrows between them

Explanation:

Step1: Recall vector subtraction formula

For vectors \(\mathbf{u}=\langle u_1, u_2
angle\) and \(\mathbf{v}=\langle v_1, v_2
angle\), \(\mathbf{u}+(-\mathbf{v})=\langle u_1 - v_1, u_2 - v_2
angle\), and the magnitude of a vector \(\langle x, y
angle\) is \(\sqrt{x^2 + y^2}\).

Step2: Calculate \( \mathbf{a}+(-\mathbf{b}) \)

\(\mathbf{a}=\langle - 3,5
angle\), \(\mathbf{b}=\langle 2,-1
angle\), so \(\mathbf{a}+(-\mathbf{b})=\langle - 3-2,5 - (-1)
angle=\langle - 5,6
angle\). Magnitude: \(\sqrt{(-5)^2 + 6^2}=\sqrt{25 + 36}=\sqrt{61}\approx7.81\).

Step3: Calculate \( \mathbf{c}+(-\mathbf{d}) \)

\(\mathbf{c}=\langle 2,7
angle\), \(\mathbf{d}=\langle 3,5
angle\), so \(\mathbf{c}+(-\mathbf{d})=\langle 2 - 3,7 - 5
angle=\langle - 1,2
angle\). Magnitude: \(\sqrt{(-1)^2+2^2}=\sqrt{1 + 4}=\sqrt{5}\approx2.24\).

Step4: Calculate \( \mathbf{e}+(-\mathbf{f}) \)

\(\mathbf{e}=\langle 4,1
angle\), \(\mathbf{f}=\langle 7,4
angle\), so \(\mathbf{e}+(-\mathbf{f})=\langle 4 - 7,1 - 4
angle=\langle - 3,-3
angle\). Magnitude: \(\sqrt{(-3)^2+(-3)^2}=\sqrt{9 + 9}=\sqrt{18}\approx4.24\).

Step5: Calculate \( \mathbf{g}+(-\mathbf{h}) \)

\(\mathbf{g}=\langle 3,-3
angle\), \(\mathbf{h}=\langle 5,-1
angle\), so \(\mathbf{g}+(-\mathbf{h})=\langle 3 - 5,-3 - (-1)
angle=\langle - 2,-2
angle\). Magnitude: \(\sqrt{(-2)^2+(-2)^2}=\sqrt{4 + 4}=\sqrt{8}\approx2.83\).

Step6: Calculate \( \mathbf{m}+(-\mathbf{n}) \)

\(\mathbf{m}=\langle 2,-5
angle\), \(\mathbf{n}=\langle - 2,4
angle\), so \(\mathbf{m}+(-\mathbf{n})=\langle 2-(-2),-5 - 4
angle=\langle 4,-9
angle\). Magnitude: \(\sqrt{4^2+(-9)^2}=\sqrt{16 + 81}=\sqrt{97}\approx9.85\).

Step7: Calculate \( \mathbf{o}+(-\mathbf{p}) \)

\(\mathbf{o}=\langle - 1,-5
angle\), \(\mathbf{p}=\langle 3,-8
angle\), so \(\mathbf{o}+(-\mathbf{p})=\langle - 1-3,-5 - (-8)
angle=\langle - 4,3
angle\). Magnitude: \(\sqrt{(-4)^2 + 3^2}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step8: Order by magnitude

Sort the magnitudes: \(\sqrt{5}\approx2.24\) (\(\mathbf{c}+(-\mathbf{d})\)) \(<\sqrt{8}\approx2.83\) (\(\mathbf{g}+(-\mathbf{h})\)) \(<\sqrt{18}\approx4.24\) (\(\mathbf{e}+(-\mathbf{f})\)) \(<5\) (\(\mathbf{o}+(-\mathbf{p})\)) \(<\sqrt{61}\approx7.81\) (\(\mathbf{a}+(-\mathbf{b})\)) \(<\sqrt{97}\approx9.85\) (\(\mathbf{m}+(-\mathbf{n})\)).

Answer:

1. \( \mathbf{c}+(-\mathbf{d}) \), where \( \mathbf{c}=\langle 2,7

angle \) and \( \mathbf{d}=\langle 3,5
angle \)

1. \( \mathbf{g}+(-\mathbf{h}) \), where \( \mathbf{g}=\langle 3,-3

angle \) and \( \mathbf{h}=\langle 5,-1
angle \)

1. \( \mathbf{e}+(-\mathbf{f}) \), where \( \mathbf{e}=\langle 4,1

angle \) and \( \mathbf{f}=\langle 7,4
angle \)

1. \( \mathbf{o}+(-\mathbf{p}) \), where \( \mathbf{o}=\langle - 1,-5

angle \) and \( \mathbf{p}=\langle 3,-8
angle \)

1. \( \mathbf{a}+(-\mathbf{b}) \), where \( \mathbf{a}=\langle - 3,5

angle \) and \( \mathbf{b}=\langle 2,-1
angle \)

1. \( \mathbf{m}+(-\mathbf{n}) \), where \( \mathbf{m}=\langle 2,-5

angle \) and \( \mathbf{n}=\langle - 2,4
angle \)