document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179249>Question 1179249</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find value of the folllwing:<BR>\n" );
document.write( "(a) (a. b) + (a x b)²<BR>\n" );
document.write( "(b) (a. b) + (a x 2b)²<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note: Here . means dot product and x means cross or vector product. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850251 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850251\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve these vector operations:\r<BR>\n" );
document.write( "\n" );
document.write( "**(a) (a · b) + |a × b|²**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Dot Product (a · b):**<BR>\n" );
document.write( "    * a · b = |a||b|cos(θ), where θ is the angle between vectors a and b.<BR>\n" );
document.write( "* **Cross Product (a × b):**<BR>\n" );
document.write( "    * |a × b| = |a||b|sin(θ)<BR>\n" );
document.write( "    * |a × b|² = |a|²|b|²sin²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "Now, let's substitute these into the given expression:\r<BR>\n" );
document.write( "\n" );
document.write( "(a · b) + |a × b|² = |a||b|cos(θ) + |a|²|b|²sin²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "However, we can simplify this further using a trigonometric identity: sin²(θ) + cos²(θ) = 1.\r<BR>\n" );
document.write( "\n" );
document.write( "* |a · b|² = |a|²|b|²cos²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore,\r<BR>\n" );
document.write( "\n" );
document.write( "(a · b)² + |a × b|² = |a|²|b|²cos²(θ) + |a|²|b|²sin²(θ)<BR>\n" );
document.write( "= |a|²|b|²(cos²(θ) + sin²(θ))<BR>\n" );
document.write( "= |a|²|b|²(1)<BR>\n" );
document.write( "= |a|²|b|²\r<BR>\n" );
document.write( "\n" );
document.write( "So, the original expression is not (a.b)^2 + |axb|^2 , but (a.b) + |axb|^2.<BR>\n" );
document.write( "Thus,\r<BR>\n" );
document.write( "\n" );
document.write( "(a · b) + |a × b|² = |a||b|cos(θ) + |a|²|b|²sin²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "This cannot be further simplified without knowing the vectors a and b.\r<BR>\n" );
document.write( "\n" );
document.write( "**(b) (a · b) + |a × 2b|²**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Dot Product (a · b):**<BR>\n" );
document.write( "    * a · b = |a||b|cos(θ)<BR>\n" );
document.write( "* **Cross Product (a × 2b):**<BR>\n" );
document.write( "    * a × 2b = 2(a × b)<BR>\n" );
document.write( "    * |a × 2b| = 2|a × b| = 2|a||b|sin(θ)<BR>\n" );
document.write( "    * |a × 2b|² = 4|a|²|b|²sin²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "Now, substitute these into the given expression:\r<BR>\n" );
document.write( "\n" );
document.write( "(a · b) + |a × 2b|² = |a||b|cos(θ) + 4|a|²|b|²sin²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "Again, this expression cannot be further simplified without knowing the vectors a and b.\r<BR>\n" );
document.write( "\n" );
document.write( "**Summary:**\r<BR>\n" );
document.write( "\n" );
document.write( "* (a · b) + |a × b|² = |a||b|cos(θ) + |a|²|b|²sin²(θ)<BR>\n" );
document.write( "* (a · b) + |a × 2b|² = |a||b|cos(θ) + 4|a|²|b|²sin²(θ)\r<BR>\n" );
document.write( "\n" );
document.write( "If the question had been (a.b)^2 + |axb|^2 , then the answer would be |a|^2|b|^2.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );