Answer to Question #148578 in Analytic Geometry for Jyotiramay Rout

Question #148578

Denote by a, b and c the column vectors a = (1 2 3), b = (-2 1 -3), c = (-2 -1 1) Calculate 2a - 5b, 2a- 5b +c, a'.b,

Expert's answer

"a=\\begin{pmatrix}\n 1 \\\\\n 2 \\\\\n3\n\\end{pmatrix}, b=\\begin{pmatrix}\n -2 \\\\\n 1 \\\\\n-3\n\\end{pmatrix}, c=\\begin{pmatrix}\n -2 \\\\\n -1 \\\\\n1\n\\end{pmatrix}"

"2a-5b=2\\begin{pmatrix}\n 1 \\\\\n 2 \\\\\n3\n\\end{pmatrix}-5\\begin{pmatrix}\n -2 \\\\\n 1 \\\\\n-3\n\\end{pmatrix}=\\begin{pmatrix}\n 2(1)-5(-2) \\\\\n 2(2)-5(1) \\\\\n2(3)-5(-3)\n\\end{pmatrix}"

"=\\begin{pmatrix}\n 12 \\\\\n -1 \\\\\n21\n\\end{pmatrix}"

"2a-5b+c=2\\begin{pmatrix}\n 1 \\\\\n 2 \\\\\n3\n\\end{pmatrix}-5\\begin{pmatrix}\n -2 \\\\\n 1 \\\\\n-3\n\\end{pmatrix}+\\begin{pmatrix}\n -2 \\\\\n -1 \\\\\n1\n\\end{pmatrix}"

"=\\begin{pmatrix}\n 12 \\\\\n -1 \\\\\n21\n\\end{pmatrix}+\\begin{pmatrix}\n -2 \\\\\n -1 \\\\\n1\n\\end{pmatrix}=\\begin{pmatrix}\n 12-2 \\\\\n -1 -1 \\\\\n21+1\n\\end{pmatrix}=\\begin{pmatrix}\n 10 \\\\\n -2 \\\\\n22\n\\end{pmatrix}"

"a^T\\cdot b=\\begin{pmatrix}\n 1 & 2 & 3\n\\end{pmatrix}\\cdot\\begin{pmatrix}\n -2 \\\\\n 1 \\\\\n-3\n\\end{pmatrix}"

"=1(-2)+2(1)+3(-3)=-9"