document.write( "Question 1203668: Let e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) ∈ R 3. Find all real numbers c ∈ R such that the angle between the vectors −e1 + 2e2 + ke3 and −e1 + ke2 + 2e3 is π/2 (they are orthogonal).
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Algebra.Com's Answer #839409 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "It appears there might be a typo.
\n" ); document.write( "I think the portion Find all real numbers c ∈ R should be Find all real numbers k ∈ R\r
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\n" ); document.write( "\n" ); document.write( "e1 = (1,0,0)
\n" ); document.write( "e2 = (0,1,0)
\n" ); document.write( "e3 = (0,0,1)\r
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\n" ); document.write( "\n" ); document.write( "v1 = -1*e1 + 2*e2 + k*e3
\n" ); document.write( "v1 = -1*(1,0,0) + 2*(0,1,0) + k*(0,0,1)
\n" ); document.write( "v1 = (-1,0,0) + (0,2,0) + (0,0,k)
\n" ); document.write( "v1 = (-1+0+0, 0+2+0, 0+0+k)
\n" ); document.write( "v1 = (-1,2,k)\r
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\n" ); document.write( "\n" ); document.write( "Follow similar steps to find that
\n" ); document.write( "v2 = -1*e1 + k*e2 + 2*e3
\n" ); document.write( "v2 = (-1,k,2)\r
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\n" ); document.write( "\n" ); document.write( "We want to have vectors v1 and v2 to be orthogonal.
\n" ); document.write( "In other words, we want the vectors to be perpendicular to each other.
\n" ); document.write( "This occurs if and only if the dot product of said vectors is 0.\r
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\n" ); document.write( "\n" ); document.write( "v1 dot v2 = (-1,2,k) dot (-1,k,2)
\n" ); document.write( "v1 dot v2 = (-1)*(-1) + 2*k + k*2
\n" ); document.write( "v1 dot v2 = 1 + 2k + 2k
\n" ); document.write( "v1 dot v2 = 1 + 4k
\n" ); document.write( "1 + 4k = 0
\n" ); document.write( "4k = -1
\n" ); document.write( "k = -1/4 is the final answer
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