October 11th, 2018, 11:15 PM  #1 
Member Joined: Oct 2012 Posts: 70 Thanks: 0  Find a vector?
Let u , v a vectors of 2dim such that u=<4,2> ,v=10 the angle between u and v is 45 degree. Find a coordinate for v satisfies these conditions.

October 12th, 2018, 09:58 AM  #2  
Math Team Joined: Jul 2011 From: Texas Posts: 2,779 Thanks: 1430 
$\cos(45^\circ) = \dfrac{\vec{u} \cdot \vec{v}}{\vec{u}\cdot\vec{v}}$ $\vec{u} = <4,2>$, let $\vec{v} = <a,b>$ $\dfrac{1}{\sqrt{2}} = \dfrac{4a + ( 2)b}{\sqrt{20} \cdot 10} \implies 4a2b = 10\sqrt{10}$ also, $a^2+b^2=100$ Quote:
 
October 13th, 2018, 06:35 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,957 Thanks: 1844 
The points (0, 0), (4, 2), (6, 2), and (2, 4) are the vertices of a square whose diagonals have length 2√10. Hence the desired vector is (1/2)<6, 2> or (1/2)<2, 6>, i.e. <3, 1>, or <1, 3>. 
October 13th, 2018, 07:50 AM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,779 Thanks: 1430  
October 13th, 2018, 09:54 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,957 Thanks: 1844 
Thanks. The coordinates I gave should therefore be multiplied by √10.


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