document.write( "Question 1112739: The vectors are shown as the x and y on top of each other but since I can’t type it out like that, it’ll be separated with commas
\n" ); document.write( "The vectors a = (4,2) and b = ( k + 3, k ) are perpendicular to each other. Find the value of k. \n" ); document.write( "
Algebra.Com's Answer #727766 by ikleyn(52797)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Thank you for your question.\r\n" );
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document.write( "    The condition that two vectors  (a,b)  and  (c,d)  in a coordinate plane are perpendicular is \r\n" );
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document.write( "    that their scalar product  (so called dot-product)  a*c + b*d  is equal to zero:\r\n" );
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document.write( "               a*c + b*d = 0\r\n" );
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document.write( "In your case it means that\r\n" );
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document.write( "               4*(k+3) = 2k.\r\n" );
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document.write( "You can easily solve this simple single linear equation\r\n" );
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document.write( "               4k + 12 = 2k  ====>  2k = - 12  ====>  k = -6.\r\n" );
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\n" ); document.write( "If you want to learn more on dot-product, look into my lessons in this site\r
\n" ); document.write( "\n" ); document.write( "    - Introduction to dot-product\r
\n" ); document.write( "\n" ); document.write( "    - Formula for Dot-product of vectors in a plane via the vectors components\r
\n" ); document.write( "\n" ); document.write( "    - Dot-product of vectors in a coordinate plane and the angle between two vectors\r
\n" ); document.write( "\n" ); document.write( "    - Perpendicular vectors in a coordinate plane\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on Dot-product of vectors and the angle between two vectors \r
\n" ); document.write( "\n" ); document.write( "    - Properties of Dot-product of vectors in a coordinate plane \r
\n" ); document.write( "\n" ); document.write( "    - The formula for the angle between two vectors and the formula for cosines of the difference of two angles\r
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\n" ); document.write( "\n" ); document.write( "There are short lessons of the  \"HOW TO . . . \"  type on Dot-product:\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO find dot-product of two vectors in a plane\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO find scalar product of two vectors in a coordinate plane\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO find the angle between two vectors in a coordinate plane\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO prove that two vectors in a coordinate plane are perpendicular\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO prove that a triangle in a coordinate plane is a right triangle\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO check if a quadrilateral in a coordinate plane is a parallelogram\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO check if a quadrilateral in a coordinate plane is a rectangle\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO check if a quadrilateral in a coordinate plane is a rhombus\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO check if a quadrilateral in a coordinate plane is a square\r
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\n" ); document.write( "\n" ); document.write( "For the full list of my lessons on dot-product with short annotations see the file  OVERVIEW of lessons on Dot-product. \r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Dot-product for vectors in a coordinate plane\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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