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# Answer to Question #225910 in Chemical Engineering for Lokika

Question #225910

The unit normal to the surface x² + y² + z² = 3 at (1,1,1) is A)1/√16 (2i+2j+2k) B)1/√3(i+j+ k)

C) 1/√6(i+2j+3k)

D)1/√3(i+2j+2k)

1
2021-08-17T12:30:01-0400

"x\u00b2 + y\u00b2 + z\u00b2 = 3\\\\\nf\n(\nx\n,\ny\n,\nz\n)= x\u00b2 + y\u00b2 + z\u00b2 - 3=0"

The gradient of "f\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n)"  at point "x\n\n,\n\ny\n\n,\n\nz"  is a vector normal to the surface at this point.

The gradient is obtained as follows

"\u2207\nf\n(\nx\n,\ny\n,\nz\n)\n=\n(\nf_\nx\n,\nf_\ny\n,\nf_\nz\n)\n= 2x+2y+2z" at point (1,1,1) and the unit vector is

"=\\frac{2,2,2}{\\sqrt{2+2+2}}\\\\\n=\\frac{2}{\\sqrt{8}}, \\frac{2}{\\sqrt{8}},\\frac{2}{\\sqrt{8}}\\\\\n=\\frac{1}{\\sqrt{2}}, \\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}}"

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