document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207395>Question 1207395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The lines l and m have vector equations<BR>\n" );
document.write( "r = 2i-j+4k+s(i+j- k) and r= -2i + 2j+ k+ t(-2i+j+ k)<BR>\n" );
document.write( "respectively・<BR>\n" );
document.write( "i) Show that l and m do not intersect.\r<BR>\n" );
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document.write( "The point P lies on / and the point @ has position vector 2i - k.<BR>\n" );
document.write( "ii)Given that the line PQ is perpendicular to l, find the position vector of P.<BR>\n" );
document.write( "(iii) Verify that Q lies on m and that PQ is perpendicular to m. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847972 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847972\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **i) Show that l and m do not intersect**\r<BR>\n" );
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document.write( "* **Find the direction vectors of the lines:** <BR>\n" );
document.write( "    * Direction vector of line l: **d₁** = i + j - k<BR>\n" );
document.write( "    * Direction vector of line m: **d₂** = -2i + j + k\r<BR>\n" );
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document.write( "* **Check if the lines are parallel:**<BR>\n" );
document.write( "    * Lines are parallel if their direction vectors are scalar multiples of each other. <BR>\n" );
document.write( "    * **d₁** is not a scalar multiple of **d₂**, so the lines are not parallel.\r<BR>\n" );
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document.write( "* **Check for intersection:**<BR>\n" );
document.write( "    * To find the point of intersection, we need to find values of 's' and 't' that satisfy the following equations:<BR>\n" );
document.write( "        * 2 + s = -2 - 2t <BR>\n" );
document.write( "        * -1 + s = 2 + t <BR>\n" );
document.write( "        * 4 - s = 1 + t\r<BR>\n" );
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document.write( "    * Solving this system of equations, we find that there is no consistent solution for 's' and 't'.\r<BR>\n" );
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document.write( "* **Conclusion:** Since the lines are not parallel and do not intersect, they are **skew lines**.\r<BR>\n" );
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document.write( "**ii) Find the position vector of P**\r<BR>\n" );
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document.write( "* **Let P be the point on line l with position vector:**<BR>\n" );
document.write( "    * **r_P = 2i - j + 4k + s(i + j - k)**\r<BR>\n" );
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document.write( "* **Find the direction vector of PQ:**<BR>\n" );
document.write( "    * **PQ = OQ - OP = (2i - k) - (2i - j + 4k + s(i + j - k))**<BR>\n" );
document.write( "        * **PQ = -j - 5k - s(i + j - k)**\r<BR>\n" );
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document.write( "* **Since PQ is perpendicular to l, their dot product must be zero:**<BR>\n" );
document.write( "    * **PQ • d₁ = 0**<BR>\n" );
document.write( "        * (-j - 5k - s(i + j - k)) • (i + j - k) = 0<BR>\n" );
document.write( "        * -s - j + k - s(i + j - k) = 0 <BR>\n" );
document.write( "        * -s - j + k - s - s = 0 <BR>\n" );
document.write( "        * -3s - j + k = 0\r<BR>\n" );
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document.write( "* **Solve for 's':**<BR>\n" );
document.write( "    * This equation implies that s = 0.\r<BR>\n" );
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document.write( "* **Find the position vector of P:**<BR>\n" );
document.write( "    * **r_P = 2i - j + 4k + 0(i + j - k)**<BR>\n" );
document.write( "        * **r_P = 2i - j + 4k**\r<BR>\n" );
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document.write( "**iii) Verify that Q lies on m and that PQ is perpendicular to m**\r<BR>\n" );
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document.write( "* **Check if Q lies on line m:**<BR>\n" );
document.write( "    * The position vector of Q is 2i - k. <BR>\n" );
document.write( "    * We need to find a value of 't' such that:<BR>\n" );
document.write( "        * 2i - k = -2i + 2j + k + t(-2i + j + k)<BR>\n" );
document.write( "        * 4i - 2j - 2k = t(-2i + j + k)\r<BR>\n" );
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document.write( "    * Comparing coefficients, we can see that t = -2 satisfies the equation. \r<BR>\n" );
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document.write( "* **Check if PQ is perpendicular to m:**<BR>\n" );
document.write( "    * **PQ = -j - 5k** (from part ii)<BR>\n" );
document.write( "    * **d₂ = -2i + j + k** (direction vector of line m)<BR>\n" );
document.write( "    * **PQ • d₂ = (-j - 5k) • (-2i + j + k) = 0 - 1 - 5 = -6**<BR>\n" );
document.write( "        * Since the dot product of PQ and d₂ is not zero, PQ is **not** perpendicular to m.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* There is an error in the problem statement or the calculations. <BR>\n" );
document.write( "* If Q lies on line m, then PQ should be perpendicular to m. \r<BR>\n" );
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document.write( "**Please double-check the given information and the calculations in part ii) to ensure accuracy.**<BR>\n" );
document.write( " </SPAN>\n" );
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