document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166411>Question 1166411</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two vectors, V1 and V2 whose sum is <-2,0>, where V1 is parallel to <0,2> while V2 is perpendicular to <0,2>.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852759 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852759\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem can be solved by setting up a system of linear equations based on the properties of parallel and perpendicular vectors in the plane.\r<BR>\n" );
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document.write( "The two vectors are:<BR>\n" );
document.write( "* $\mathbf{V}_1 = \langle 0, 2 \rangle$<BR>\n" );
document.write( "* $\mathbf{V}_2 = \langle -2, 0 \rangle$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 📐 Setting Up the Equations\r<BR>\n" );
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document.write( "Let the two unknown vectors be:<BR>\n" );
document.write( "$$\mathbf{V}_1 = \langle a, b \rangle$$<BR>\n" );
document.write( "$$\mathbf{V}_2 = \langle c, d \rangle$$\r<BR>\n" );
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document.write( "### 1. The Sum Condition\r<BR>\n" );
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document.write( "The sum of the two vectors is $\langle -2, 0 \rangle$:<BR>\n" );
document.write( "$$\mathbf{V}_1 + \mathbf{V}_2 = \langle a + c, b + d \rangle = \langle -2, 0 \rangle$$<BR>\n" );
document.write( "This gives us two scalar equations:<BR>\n" );
document.write( "1.  $a + c = -2$<BR>\n" );
document.write( "2.  $b + d = 0$\r<BR>\n" );
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document.write( "### 2. The Parallel Condition ($\mathbf{V}_1$ is parallel to $\langle 0, 2 \rangle$)\r<BR>\n" );
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document.write( "If $\mathbf{V}_1$ is parallel to $\langle 0, 2 \rangle$, then $\mathbf{V}_1$ must be a scalar multiple ($k$) of $\langle 0, 2 \rangle$:<BR>\n" );
document.write( "$$\mathbf{V}_1 = k \langle 0, 2 \rangle = \langle 0, 2k \rangle$$<BR>\n" );
document.write( "Comparing this to $\mathbf{V}_1 = \langle a, b \rangle$:<BR>\n" );
document.write( "3.  $a = 0$<BR>\n" );
document.write( "4.  $b = 2k$\r<BR>\n" );
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document.write( "### 3. The Perpendicular Condition ($\mathbf{V}_2$ is perpendicular to $\langle 0, 2 \rangle$)\r<BR>\n" );
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document.write( "Two vectors are perpendicular if their **dot product** is zero \r<BR>\n" );
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document.write( "[Image of two perpendicular vectors showing the dot product is zero]<BR>\n" );
document.write( ".<BR>\n" );
document.write( "$$\mathbf{V}_2 \cdot \langle 0, 2 \rangle = 0$$<BR>\n" );
document.write( "$$\langle c, d \rangle \cdot \langle 0, 2 \rangle = (c)(0) + (d)(2) = 0$$<BR>\n" );
document.write( "$$2d = 0$$<BR>\n" );
document.write( "5.  $d = 0$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 🧠 Solving the System\r<BR>\n" );
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document.write( "Now we substitute the determined values back into the sum conditions:\r<BR>\n" );
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document.write( "* From Equation 3, we know **$a = 0$**.<BR>\n" );
document.write( "* From Equation 5, we know **$d = 0$**.\r<BR>\n" );
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document.write( "1.  **Use $a=0$ in the first sum equation:**<BR>\n" );
document.write( "    $$a + c = -2$$<BR>\n" );
document.write( "    $$0 + c = -2$$<BR>\n" );
document.write( "    $$\mathbf{c = -2}$$\r<BR>\n" );
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document.write( "2.  **Use $d=0$ in the second sum equation:**<BR>\n" );
document.write( "    $$b + d = 0$$<BR>\n" );
document.write( "    $$b + 0 = 0$$<BR>\n" );
document.write( "    $$\mathbf{b = 0}$$\r<BR>\n" );
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document.write( "### Final Vectors\r<BR>\n" );
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document.write( "Substitute the values back into the vector definitions:<BR>\n" );
document.write( "$$\mathbf{V}_1 = \langle a, b \rangle = \langle 0, 0 \rangle$$<BR>\n" );
document.write( "$$\mathbf{V}_2 = \langle c, d \rangle = \langle -2, 0 \rangle$$\r<BR>\n" );
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document.write( "---<BR>\n" );
document.write( "**Wait! Let's re-read the perpendicular condition.**\r<BR>\n" );
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document.write( "The vector $\mathbf{V}_1 = \langle 0, 0 \rangle$ is parallel to any vector. $\mathbf{V}_2 = \langle -2, 0 \rangle$ is perpendicular to $\langle 0, 2 \rangle$ because $\langle -2, 0 \rangle \cdot \langle 0, 2 \rangle = 0$.\r<BR>\n" );
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document.write( "However, there is a fundamental issue with $\mathbf{V}_1 = \langle 0, 0 \rangle$ being the unique solution for the parallel vector. A vector parallel to $\langle 0, 2 \rangle$ **must only have a vertical component**.\r<BR>\n" );
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document.write( "Let $\mathbf{P} = \langle 0, 2 \rangle$.<BR>\n" );
document.write( "The space of vectors parallel to $\mathbf{P}$ is the set $S_{\parallel} = \{\langle 0, y \rangle\}$.<BR>\n" );
document.write( "The space of vectors perpendicular to $\mathbf{P}$ is the set $S_{\perp} = \{\langle x, 0 \rangle\}$ (since $2y=0$ implies $y=0$ in the dot product).\r<BR>\n" );
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document.write( "Let $\mathbf{V}_1 = \langle 0, y_1 \rangle$ and $\mathbf{V}_2 = \langle x_2, 0 \rangle$.\r<BR>\n" );
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document.write( "Their sum must be $\langle -2, 0 \rangle$:<BR>\n" );
document.write( "$$\mathbf{V}_1 + \mathbf{V}_2 = \langle 0 + x_2, y_1 + 0 \rangle = \langle -2, 0 \rangle$$\r<BR>\n" );
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document.write( "By comparison of components:<BR>\n" );
document.write( "* $x_2 = -2$<BR>\n" );
document.write( "* $y_1 = 0$\r<BR>\n" );
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document.write( "This confirms the initial result:<BR>\n" );
document.write( "$$\mathbf{V}_1 = \langle 0, 0 \rangle$$<BR>\n" );
document.write( "$$\mathbf{V}_2 = \langle -2, 0 \rangle$$ </SPAN>\n" );
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