document.write( "Question 1006183: find two vectors v1 and v2 whose sum is <3,-2> where v1 is parallel to <-4,5> while v2 is perpendicular to <-4,5>\r
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Algebra.Com's Answer #622341 by AnlytcPhil(1807)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I don't understand why the other tutor thought it mattered that\r\n" );
document.write( "< -4,5 > was mentioned twice.  Here is the picture of how it\r\n" );
document.write( "is easily possible.  I'll put all the vectors with their tails \r\n" );
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document.write( "Your problem was:\r\n" );
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\n" ); document.write( "find two vectors v1 and v2 whose sum is <3,-2> where v1
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document.write( "As you can see from the graph, it certainly looks possible for v1\r\n" );
document.write( "to be parallel to <-4,5> and v2 to be perpendicular to it.\r\n" );
document.write( "And also the green lines show that the sum of v1 and v2 can\r\n" );
document.write( "be <3,-2>. \r\n" );
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document.write( "[I hope it doesn't bother you that v1 and <-4,5> in the graph are \r\n" );
document.write( "both on the same line and I said they are parallel.  Remember that \r\n" );
document.write( "vectors can be moved anywhere, and so vectors which are multiples \r\n" );
document.write( "of each other are considered to be parallel because they can be \r\n" );
document.write( "moved so that they are parallel by the usual standards of basic \r\n" );
document.write( "geometry.] \r\n" );
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document.write( "So let's calculate what we see in the above drawing:\r\n" );
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document.write( "Since v1 is parallel to < -4,5 >, there is some non-zero scalar k \r\n" );
document.write( "such that\r\n" );
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document.write( "v1 = < -4k,5k >\r\n" );
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document.write( "let v2 = < a,b >\r\n" );
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document.write( "Since v2 is perpendicular to < -4,5 > its dot product \r\n" );
document.write( "with < -4,5 > is 0, so\r\n" );
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document.write( "v2•< -4,5 > = < a,b >•< -4,5 > = -4a+5b = 0\r\n" );
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document.write( "(1)    -4a+5b = 0\r\n" );
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document.write( "Since v1 + v2 = <3,-2> \r\n" );
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document.write( "  < -4k,5k > + < a,b > = < 3, -2 >\r\n" );
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document.write( "(2)    -4k+a = 3\r\n" );
document.write( "(3)     5k+b = -2\r\n" );
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document.write( "So we have this system of 3 equations in 3 unknowns:\r\n" );
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document.write( "  \r\n" );
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document.write( "The variable \"a\" is already eliminated in the 3rd equation,\r\n" );
document.write( "so we eliminate \"a\" from the 1st and 2nd equations, by\r\n" );
document.write( "multiplying the 2nd equation by 4 and adding it to the\r\n" );
document.write( "1st equation:\r\n" );
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document.write( "-4a + 5b       =  0\r\n" );
document.write( " 4a      + 16k = 12\r\n" );
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document.write( "      5b + 16k = 12\r\n" );
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document.write( "Now we have the system of 2 equations in 2 unknowns:\r\n" );
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document.write( "Eliminate b by multiplying the second equation through \r\n" );
document.write( "by -5 and adding it to the first equation:\r\n" );
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document.write( " 5b - 16k = 12\r\n" );
document.write( "-5b - 25k = 10\r\n" );
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document.write( "     -41k = 22\r\n" );
document.write( "        k = \"-22%2F41\"\r\n" );
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document.write( "Substituting in \r\n" );
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document.write( "(2)    -4k+a = 3\r\n" );
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document.write( "We get:\r\n" );
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document.write( "      \"-4%28-22%2F41%29%2Ba=3\"\r\n" );
document.write( "      \"88%2F41%2Ba=3\"\r\n" );
document.write( "      \"a=3-88%2F41\"\r\n" );
document.write( "      \"a=123%2F41-88%2F41\"\r\n" );
document.write( "      \"a=35%2F41\"\r\n" );
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document.write( "Substituting in   \r\n" );
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document.write( "(3)     5k+b = -2\r\n" );
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document.write( "We get:\r\n" );
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document.write( "      \"5%28-22%2F41%29%2Bb=-2\"\r\n" );
document.write( "      \"-110%2F41%2Bb=-2\"\r\n" );
document.write( "      \"b=-2%2B110%2F41\"\r\n" );
document.write( "      \"b=-82%2F41%2B110%2F41\"\r\n" );
document.write( "      \"b=28%2F41\"\r\n" );
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document.write( "So the answer is:\r\n" );
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document.write( "v1 = < -4k,5k > =  = \"matrix%281%2C5%2C%22%22%3C%22%22%2C88%2F41%2C%22%2C%22%2C-110%2F41%2C%22%22%3E%22%22%29\"\r\n" );
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document.write( "v2 = < a,b > = \"matrix%281%2C5%2C%22%22%3C%22%22%2C35%2F41%2C%22%2C%22%2C28%2F41%2C%22%22%3E%22%22%29\"\r\n" );
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document.write( "That's a pretty messy answer, but it's correct, and it looks\r\n" );
document.write( "like it agrees with the drawing for v1 and v2 above. \r\n" );
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document.write( "Edwin
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