<PRE><B>Question 1187474</B>
&lt;pre&gt;
I will use &lt;b&gt;bold&lt;/b&gt; &lt;b&gt;v&lt;/b&gt;, &lt;b&gt;u&lt;/b&gt;, and &lt;b&gt;w&lt;/b&gt; to denote vectors, and
notation like {{{u[x]}}} to denote components of a vector (here x component of vector &lt;b&gt;u&lt;/b&gt;)  

Hopefully it renders properly.


&lt;b&gt;u&lt;/b&gt; = 2&lt;b&gt;i&lt;/b&gt; - 3&lt;b&gt;j&lt;/b&gt;           (1)

From &lt;b&gt;v&lt;/b&gt; || &lt;b&gt;u&lt;/b&gt;: &lt;b&gt;v&lt;/b&gt; = 2c&lt;b&gt;i&lt;/b&gt;-3c&lt;b&gt;j&lt;/b&gt;   (2)
   where  c  is some scaler value.


From &lt;b&gt;w&lt;/b&gt; perpendicular to &lt;b&gt;u&lt;/b&gt;:  &lt;b&gt;w&lt;/b&gt;&amp;#x2219;&lt;b&gt;u&lt;/b&gt; = 0
   which translates to
         2{{{w[x]}}} - 3{{{w[y]}}} = 0     (3)

We also want:  &lt;b&gt;w&lt;/b&gt; + &lt;b&gt;v&lt;/b&gt; = 4&lt;b&gt;i&lt;/b&gt; + 4&lt;b&gt;j&lt;/b&gt;    (4) 

(3) ==&gt;  {{{ w[y] = (2/3)w[x] }}}

Now we can write, using (3) and (4):
 (5)   2c + {{{w[x]}}} = 4
 {6)  -3c + {{{(2/3)w[x]}}} = 4

Solving this for c gives  {{{c=-4/13}}}  which tells us via (2)
that &lt;b&gt;v&lt;/b&gt; = (-8/13)&lt;b&gt;i&lt;/b&gt; + (12/13)&lt;b&gt;j&lt;/b&gt;


Now you can use (5) and (6) to solve for {{{w[x]}}} or plug  c into 
(4) and solve for {{{w[x]}}} and {{{w[y]}}} (I did it the latter way):

 &lt;b&gt;w&lt;/b&gt; = (60/13)&lt;b&gt;i&lt;/b&gt; + (40/13)&lt;b&gt;j&lt;/b&gt;



Check:
I checked everything but you should do so as an exercise.
1.  Check that &lt;b&gt;w&lt;/b&gt; + &lt;b&gt;v&lt;/b&gt; = 4&lt;b&gt;i&lt;/b&gt; + 4&lt;b&gt;j&lt;/b&gt;
2.  Check that &lt;b&gt;w&lt;/b&gt;&amp;#x2219;&lt;b&gt;u&lt;/b&gt; = 0  ({{{w[x]u[x] + w[y]u[y] = 0 }}})
3.  Check that |&lt;b&gt;v&lt;/b&gt;&amp;#x2219;&lt;b&gt;u&lt;/b&gt;| = |&lt;b&gt;v&lt;/b&gt;||&lt;b&gt;u&lt;/b&gt;|

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