document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163824>Question 1163824</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let vectors u and v form an angle θ when placed tail-to-tail, and let w be the vector projection of v onto u.<BR>\n" );
document.write( "(a)Assume that θ is acute. Notice that w points in the same direction as u. Find |w|, then<BR>\n" );
document.write( "show that w = |v| u•v 1 u, which simplifies to just w = u•v u. |u||v| |u| u•u<BR>\n" );
document.write( "(b)If θ is obtuse, do w and u point in the same direction? Does formula (a) still work? By the way, the notation projuv is sometimes used for the vector projection of v onto u. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854369 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854369\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ### (a) Analysis of the Acute Case\r<BR>\n" );
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document.write( "When $\theta$ is **acute** ($0 \le \theta < 90^\circ$), the vector projection $\mathbf{w}$ points in the same direction as $\mathbf{u}$.\r<BR>\n" );
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document.write( "**1. Finding $|\mathbf{w}|$:**<BR>\n" );
document.write( "By forming a right triangle where $\mathbf{v}$ is the hypotenuse and $\mathbf{w}$ is the adjacent side, we use basic trigonometry:<BR>\n" );
document.write( "$$|\mathbf{w}| = |\mathbf{v}| \cos \theta$$\r<BR>\n" );
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document.write( "**2. Showing the Formula:**<BR>\n" );
document.write( "To find the vector $\mathbf{w}$, we multiply its magnitude by a unit vector in the direction of $\mathbf{u}$. The unit vector for $\mathbf{u}$ is $\frac{\mathbf{u}}{|\mathbf{u}|}$.<BR>\n" );
document.write( "$$\mathbf{w} = (|\mathbf{v}| \cos \theta) \frac{\mathbf{u}}{|\mathbf{u}|}$$\r<BR>\n" );
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document.write( "From the geometric definition of the dot product, $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta$. We can solve for $\cos \theta$:<BR>\n" );
document.write( "$$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$$\r<BR>\n" );
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document.write( "Substitute this back into the equation for $\mathbf{w}$:<BR>\n" );
document.write( "$$\mathbf{w} = |\mathbf{v}| \left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|} \right) \frac{\mathbf{u}}{|\mathbf{u}|}$$\r<BR>\n" );
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document.write( "**3. Simplification:**<BR>\n" );
document.write( "The $|\mathbf{v}|$ terms cancel out:<BR>\n" );
document.write( "$$\mathbf{w} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}|^2} \mathbf{u}$$\r<BR>\n" );
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document.write( "Since $|\mathbf{u}|^2 = \mathbf{u} \cdot \mathbf{u}$, the formula simplifies to:<BR>\n" );
document.write( "$$\mathbf{w} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{u} \cdot \mathbf{u}} \right) \mathbf{u}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### (b) Analysis of the Obtuse Case\r<BR>\n" );
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document.write( "**1. Do $\mathbf{w}$ and $\mathbf{u}$ point in the same direction?**<BR>\n" );
document.write( "**No.** If $\theta$ is **obtuse** ($90^\circ < \theta \le 180^\circ$), the vector $\mathbf{v}$ points \"away\" from the direction of $\mathbf{u}$. Consequently, the projection $\mathbf{w}$ will point in the **opposite direction** of $\mathbf{u}$. \r<BR>\n" );
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document.write( "**2. Does the formula from (a) still work?**<BR>\n" );
document.write( "**Yes, the formula is robust.** Here is why:<BR>\n" );
document.write( "* When $\theta$ is obtuse, $\cos \theta$ is **negative**. <BR>\n" );
document.write( "* This makes the dot product $\mathbf{u} \cdot \mathbf{v}$ negative. <BR>\n" );
document.write( "* In the formula $\mathbf{w} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{u} \cdot \mathbf{u}} \right) \mathbf{u}$, the scalar multiplier becomes negative.<BR>\n" );
document.write( "* A negative scalar multiplied by vector $\mathbf{u}$ automatically flips its direction, correctly resulting in a vector projection $\mathbf{w}$ that points opposite to $\mathbf{u}$.\r<BR>\n" );
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document.write( "Thus, the notation $\text{proj}_{\mathbf{u}}\mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$ is universal for any angle $\theta$ (except when $\mathbf{u}$ is the zero vector). </SPAN>\n" );
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