Question

drag each tile to the correct box.
vector ( mathbf{u} = (9, -2) ), ( mathbf{v} = (-1, 7) ), and ( mathbf{w} = (-5, -8) ). arrange the vector operations in ascending order of the magnitudes of their resultant vectors

( -\frac{1}{2}mathbf{u} + 5mathbf{v} )
( \frac{1}{6}(mathbf{u} + 2mathbf{v} - mathbf{w}) )
( \frac{5}{2}mathbf{u} - 3mathbf{w} )
( mathbf{u} - \frac{3}{2}mathbf{v} + 2mathbf{w} )
( -4mathbf{v} + \frac{1}{2}mathbf{w} )
( 3mathbf{u} - mathbf{v} - \frac{5}{2}mathbf{w} )

Explanation:

Step1: Recall vector operations

For a vector \(\mathbf{a}=(x_1,y_1)\) and \(\mathbf{b}=(x_2,y_2)\), scalar multiplication \(k\mathbf{a}=(kx_1,ky_1)\) and vector addition \(\mathbf{a}+\mathbf{b}=(x_1 + x_2,y_1 + y_2)\). The magnitude of a vector \(\mathbf{v}=(x,y)\) is \(|\mathbf{v}|=\sqrt{x^{2}+y^{2}}\).

Step2: Calculate \(-\frac{1}{2}\mathbf{u}+5\mathbf{v}\)

Given \(\mathbf{u}=(9,-2)\), \(\mathbf{v}=(-1,7)\)
\(-\frac{1}{2}\mathbf{u}=(-\frac{9}{2},1)\), \(5\mathbf{v}=(-5,35)\)
\(-\frac{1}{2}\mathbf{u}+5\mathbf{v}=(-\frac{9}{2}-5,1 + 35)=(-\frac{19}{2},36)\)
Magnitude: \(\sqrt{(-\frac{19}{2})^{2}+36^{2}}=\sqrt{\frac{361}{4}+1296}=\sqrt{\frac{361 + 5184}{4}}=\sqrt{\frac{5545}{4}}\approx\sqrt{1386.25}\approx37.23\)

Step3: Calculate \(\frac{1}{6}(\mathbf{u}+2\mathbf{v}-\mathbf{w})\)

\(\mathbf{w}=(-5,-8)\)
\(\mathbf{u}+2\mathbf{v}-\mathbf{w}=(9,-2)+2(-1,7)-(-5,-8)=(9-2 + 5,-2+14 + 8)=(12,20)\)
\(\frac{1}{6}(\mathbf{u}+2\mathbf{v}-\mathbf{w})=(2,\frac{10}{3})\)
Magnitude: \(\sqrt{2^{2}+(\frac{10}{3})^{2}}=\sqrt{4+\frac{100}{9}}=\sqrt{\frac{36 + 100}{9}}=\sqrt{\frac{136}{9}}\approx\sqrt{15.11}\approx3.89\)

Step4: Calculate \(\frac{5}{2}\mathbf{u}-3\mathbf{w}\)

\(\frac{5}{2}\mathbf{u}=(\frac{45}{2},-5)\), \(-3\mathbf{w}=(15,24)\)
\(\frac{5}{2}\mathbf{u}-3\mathbf{w}=(\frac{45}{2}+15,-5 + 24)=(\frac{75}{2},19)\)
Magnitude: \(\sqrt{(\frac{75}{2})^{2}+19^{2}}=\sqrt{\frac{5625}{4}+361}=\sqrt{\frac{5625+1444}{4}}=\sqrt{\frac{7069}{4}}\approx\sqrt{1767.25}\approx42.04\)

Step5: Calculate \(\mathbf{u}-\frac{3}{2}\mathbf{v}+2\mathbf{w}\)

\(-\frac{3}{2}\mathbf{v}=(\frac{3}{2},- \frac{21}{2})\), \(2\mathbf{w}=(-10,-16)\)
\(\mathbf{u}-\frac{3}{2}\mathbf{v}+2\mathbf{w}=(9+\frac{3}{2}-10,-2-\frac{21}{2}-16)=(-\frac{1}{2},-\frac{57}{2})\)
Magnitude: \(\sqrt{(-\frac{1}{2})^{2}+(-\frac{57}{2})^{2}}=\sqrt{\frac{1 + 3249}{4}}=\sqrt{\frac{3250}{4}}\approx\sqrt{812.5}\approx28.50\)

Step6: Calculate \(-4\mathbf{v}+\frac{1}{2}\mathbf{w}\)

\(-4\mathbf{v}=(4,-28)\), \(\frac{1}{2}\mathbf{w}=(-\frac{5}{2},-4)\)
\(-4\mathbf{v}+\frac{1}{2}\mathbf{w}=(4-\frac{5}{2},-28-4)=(\frac{3}{2},-32)\)
Magnitude: \(\sqrt{(\frac{3}{2})^{2}+(-32)^{2}}=\sqrt{\frac{9}{4}+1024}=\sqrt{\frac{9 + 4096}{4}}=\sqrt{\frac{4105}{4}}\approx\sqrt{1026.25}\approx32.04\)

Step7: Calculate \(3\mathbf{u}-\mathbf{v}-\frac{5}{2}\mathbf{w}\)

\(3\mathbf{u}=(27,-6)\), \(-\mathbf{v}=(1,-7)\), \(-\frac{5}{2}\mathbf{w}=(\frac{25}{2},20)\)
\(3\mathbf{u}-\mathbf{v}-\frac{5}{2}\mathbf{w}=(27 + 1+\frac{25}{2},-6-7 + 20)=(\frac{54 + 2+25}{2},7)=(\frac{81}{2},7)\)
Magnitude: \(\sqrt{(\frac{81}{2})^{2}+7^{2}}=\sqrt{\frac{6561}{4}+49}=\sqrt{\frac{6561+196}{4}}=\sqrt{\frac{6757}{4}}\approx\sqrt{1689.25}\approx41.10\)

Step8: Order by magnitude

From smallest to largest magnitude:
\(\frac{1}{6}(\mathbf{u}+2\mathbf{v}-\mathbf{w})\) (≈3.89), \(\mathbf{u}-\frac{3}{2}\mathbf{v}+2\mathbf{w}\) (≈28.50), \(-4\mathbf{v}+\frac{1}{2}\mathbf{w}\) (≈32.04), \(-\frac{1}{2}\mathbf{u}+5\mathbf{v}\) (≈37.23), \(3\mathbf{u}-\mathbf{v}-\frac{5}{2}\mathbf{w}\) (≈41.10), \(\frac{5}{2}\mathbf{u}-3\mathbf{w}\) (≈42.04)

Answer:

The order from smallest to largest magnitude of resultant vectors is:
\(\boldsymbol{\frac{1}{6}(\mathbf{u}+2\mathbf{v}-\mathbf{w})}\), \(\boldsymbol{\mathbf{u}-\frac{3}{2}\mathbf{v}+2\mathbf{w}}\), \(\boldsymbol{-4\mathbf{v}+\frac{1}{2}\mathbf{w}}\), \(\boldsymbol{-\frac{1}{2}\mathbf{u}+5\mathbf{v}}\), \(\boldsymbol{3\mathbf{u}-\mathbf{v}-\frac{5}{2}\mathbf{w}}\), \(\boldsymbol{\frac{5}{2}\mathbf{u}-3\mathbf{w}}\)