Finding the Distance Between Two Points Using Vectors

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Encyclopedia>Numbers and Quantities>Vectors>Two Dimensions>Introduction to Vectors>Finding the Distance Between Two Points Using Vectors

A vector from (8, 12) to (16, 18)

The distance between two points $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ is the length of the vector $\vec{P Q}$ . This formula finds the distance between the two points (the length of the vector) for you:

Formula

The Distance Between Two Points

| \vec{P Q} | = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

Example 1

Find the distance between $A = (8, 12)$ and $B = (16, 18)$ .

\begin{array}{l} | \vec{A B} | = \sqrt{} = \sqrt{8^{2} + 6^{2}} = \sqrt{64 + 36} = \sqrt{100} = 10Example 2 Decide s such that | \vec{A B} | = 4, when A = (s, 1) and B = (2, 5) Here, you can insert the numbers right into the formula and solve for s : \begin{array}{l} | \vec{A B} | = \sqrt{{(2 - s)}^{2} + {(5 - 1)}^{2}} & = 4 \\ \sqrt{{(2 - s)}^{2} + {(4)}^{2}} & = 4 \\ {(2 - s)}^{2} + 16 & = 16 \\ 4 - 4 s + s^{2} & = 0 \end{array} Solving the equation s^{2} - 4 s + 4 = 0 : s = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 4}}{2} = \frac{4}{2} = 2 . Because this is a radical equation, you need to test your answer : The left-hand side is \sqrt{{(2 - 2)}^{2} + {(5 - 1)}^{2}} = \sqrt{16} = 4, and the right-hand side is 4. As the left-hand and right-hand side are equal, s = 2 makes the length | \vec{A B} | equal to 4.Want to know more? Sign Up It's free!Previous entry What Is the Length of a Vector? Want to know more? Sign Up It's free!Finding the Distance Between Two Points Using Vectors Numbers and Quantities Numbers Arithmetic Fractions and Percentages Quantities Economy Vectors Two Dimensions Introduction to Vectors Products and Vector Calculation Three Dimensions Algebra Geometry Statistics and Probability Functions ProofsMenu Home Search ChatAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com + 47 22 150 300 Bygdøy allé 23,
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The distance between two points is the length of the vector.","images":[{"_key":"b7826bb9-bfed-5de6-bbe3-46a8168e52a2","type":"textwidth","dimension":"width","value":0.75,"url":"https://images.houseofmath.com/overleaf-assets/images/vektoravp.svg","alt":"A vector from (8, 12) to (16, 18)"}],"siblings":[{"h1":"What Is a Vector?","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/what-is-a-vector"},{"h1":"What Does Vector Coordinates Mean?","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/what-does-vector-coordinates-mean"},{"h1":"What Does Position Vector Mean?","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/what-does-position-vector-mean"},{"h1":"How Does Vector Addition Work?","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/how-does-vector-addition-work"},{"h1":"Multiplying Vector by Scalar","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/multiplying-vector-by-scalar"},{"h1":"How to Find the Vector Between Two Points","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/how-to-find-the-vector-between-two-points"},{"h1":"What Is the Length of a Vector?","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/what-is-the-length-of-a-vector"},{"h1":"Finding the Distance Between Two Points Using Vectors","path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/finding-the-distance-between-two-points-using-vectors"}],"isRootPath":false,"h1":"Finding the Distance Between Two Points Using Vectors","locale":"en","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy8xMDhkaXN0YW5jZXBvaW50cy5wbmciLCJidWNrZXQiOiJob20tZnJvbnRlbmQtYXNzZXRzIiwidmVyc2lvbiI6M30=","cssUrl":"/0e751716-e4cf-4658-ac87-c97d9167170e-en-1.css","htmlContent":" \u003c/p\u003e \u003cdiv class=\"center\"  \u003e \u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/vektoravp.svg\" alt=\"A vector from (8, 12) to (16, 18)\" width=\"451\" height=\"451\"\u003e\u003c/p\u003e\u003c/div\u003e \u003cp class=\"indent\" \u003e     The distance between two points \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eP\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub  \u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub  \u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eQ\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub  \u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub  \u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e is the \u003ca href=\"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/what-is-the-length-of-a-vector\"\u003elength of the vector\u003c/a\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eP\u003c/mi\u003e\u003cmi  \u003eQ\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e\to\u003c/mo\u003e\u003c/mover\u003e\u003c/math\u003e. This formula finds the distance between the two points (the length of the vector) for you:                                                                                                                                                                                                                                          \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooformula\" id=\"tcolobox-258\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Formula\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"formula-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eThe\u003c/span\u003e \u003cspan  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xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eA\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eand\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle 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\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e    \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                           \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-260\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eDecide\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003es\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003esuch\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ethat\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e\to\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003e,\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ewhen\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eA\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003es\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eand\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eB\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eHere, you can insert the numbers right into the formula and solve for \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003es\u003c/mi\u003e\u003c/math type=\"divide\"\u003e: \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                                \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e\to\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmi  \u003es\u003c/mi\u003e \u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e \u003c/mrow\u003e \u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e \u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmi  \u003es\u003c/mi\u003e \u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e \u003c/mrow\u003e \u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e \u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mtd\u003e                                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmi  \u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mtd\u003e                                                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003es\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003eSolving the equation \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003es\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math type=\"divide\"\u003e: \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                  \u003cmi  \u003es\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e\pm\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e\cdot\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e\cdot\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e              \u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e             \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"punct\"\u003e.\u003c/mo\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e \u003cp class=\"noindent\" \u003eBecause this is a \u003ca href=\"/encyclopedia/algebra/equations-and-inequalities/equations/radical-equations\"\u003eradical equation\u003c/a\u003e, you need to \u003ca href=\"/encyclopedia/algebra/equations-and-inequalities/equations/techniques/how-to-check-solutions-to-an-equation\"\u003etest your answer\u003c/a type=\"divide\"\u003e: \u003c/p\u003e\u003cp class=\"noindent\" \u003eThe left-hand side is \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                  \u003cmsqrt\u003e\u003cmrow\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e-\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e \u003cp class=\"noindent\" \u003eand the right-hand side is 4. As the left-hand and right-hand side are equal, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003es\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/math\u003e makes the length \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e|\u003c/mo\u003e\u003cmrow\u003e\u003cmover  accent=\"true\"\u003e\u003cmrow  \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo accent=\"true\"\u003e\to\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e equal to 4.  \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                               \u003c/div\u003e                                                                                                                            \u003cp class=\"indent\" \u003e     ","title":"Finding the Distance Between Two Points Using Vectors","type":"encyclopedia","image":{"type":"textwidth","dimension":"width","value":0.75,"url":"https://images.houseofmath.com/overleaf-assets/images/vektoravp.svg","alt":"A vector from (8, 12) to (16, 18)","_key":"b7826bb9-bfed-5de6-bbe3-46a8168e52a2"},"otherLocales":[],"path":"/encyclopedia/numbers-and-quantities/vectors/two-dimensions/introduction-to-vectors/finding-the-distance-between-two-points-using-vectors","breadcrumbs":[{"path":"/encyclopedia","h1":"Encyclopedia","_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c"},{"path":"/encyclopedia/numbers-and-quantities","h1":"Numbers and 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It is given by the formula:\"}],\"_type\":\"block\",\"style\":\"normal\",\"_key\":\"28c87da8b128\"},{\"markDefs\":[],\"children\":[{\"_type\":\"span\",\"marks\":[],\"text\":\"\",\"_key\":\"56b59f05c26e\"},{\"_type\":\"latex\",\"_key\":\"079101706ced\",\"body\":\" y=a(1 \pm r)\"},{\"_type\":\"span\",\"marks\":[],\"text\":\"\",\"_key\":\"55345ab5121b\"}],\"_type\":\"block\",\"style\":\"normal\",\"_key\":\"aabf3ab6d47c\"}]}]},\"name\":\"What Does the Final Amount Look like When the Growth Factor Is Equal to 1?\",\"howContent\":[{\"_type\":\"block\",\"style\":\"normal\",\"_key\":\"f42a5378cae6\",\"markDefs\":[],\"children\":[{\"_key\":\"494d669f4a800\",\"_type\":\"span\",\"marks\":[],\"text\":\"When the growth factor is exactly 1, meaning the growth rate, \"},{\"_type\":\"latex\",\"_key\":\"34dd60f8fa71\",\"body\":\"r\"},{\"_type\":\"span\",\"marks\":[],\"text\":\", is equal to 0, the final amount and initial amount are the same. 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Also known as decay rate.\",\"_key\":\"97ac7702e4720\",\"_type\":\"span\",\"marks\":[]}],\"_type\":\"block\",\"style\":\"normal\"}]}]}},{\"level\":\"7\",\"metaDescription\":\"To find the relative percentage change via growth factor, you need to find the growth or decay rate, by using the formula y =a (1 \pm r). 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