train/precalculus/1048.json

{
    "problem": "Find the vector $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}} \\mathbf{v} = \\begin{pmatrix} \\frac{38}{5} \\\\ \\frac{19}{5} \\end{pmatrix}\\]and\n\\[\\operatorname{proj}_{\\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix}} \\mathbf{v} = \\begin{pmatrix} \\frac{58}{13} \\\\ \\frac{87}{13} \\end{pmatrix}.\\]",
    "level": "Level 4",
    "type": "Precalculus",
    "solution": "Let $\\mathbf{v} = \\begin{pmatrix} x \\\\ y \\end{pmatrix}.$\n\n[asy]\nusepackage(\"amsmath\");\n\nunitsize(0.6 cm);\n\npair P, Q, V;\n\nV = (7,5);\nP = (38/5,19/5);\nQ = (58/13,87/13);\n\ndraw((-1,0)--(8,0));\ndraw((0,-1)--(0,7));\ndraw((0,0)--V,Arrow(6));\ndraw(V--P,dashed);\ndraw((-1,-1/2)--(8,4));\ndraw((0,0)--P,red,Arrow(6));\ndraw((-2/3,-1)--(2/3*7,7));\ndraw(V--Q,dashed);\ndraw((0,0)--Q,red,Arrow(6));\n\nlabel(\"$\\mathbf{v}$\", V, NE);\nlabel(\"$\\begin{pmatrix} \\frac{38}{5} \\\\ \\frac{19}{5} \\end{pmatrix}$\", P, SE);\nlabel(\"$\\begin{pmatrix} \\frac{58}{13} \\\\ \\frac{87}{13} \\end{pmatrix}$\", Q, NW);\n[/asy]\n\nThen by the properties of projections,\n\\[\\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} \\frac{38}{5} \\\\ \\frac{19}{5} \\end{pmatrix} \\right) \\cdot \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix} = 0,\\]and\n\\[\\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} \\frac{58}{13} \\\\ \\frac{87}{13} \\end{pmatrix} \\right) \\cdot \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} = 0.\\]These lead to the equations\n\\[2 \\left( x - \\frac{38}{5} \\right) + \\left( y - \\frac{19}{5} \\right) = 0\\]and\n\\[2 \\left( x - \\frac{58}{13} \\right) + 3 \\left( y - \\frac{87}{13} \\right) = 0.\\]Solving, we find $x = 7$ and $y = 5,$ so $\\mathbf{v} = \\boxed{\\begin{pmatrix} 7 \\\\ 5 \\end{pmatrix}}.$"
}