May 3rd, 2018, 04:22 AM  #1 
Newbie Joined: Aug 2017 From: Norway Posts: 10 Thanks: 0  Length of vectors
Hey! I need some help with this task: Suppose that u1,u2 is an orthonormal basis of R2, that means u1*u2=0 and ui*ui=1, i=1,2. Let y=[<y1,y2>] be a vector in R2 and v = y1u1 + y2u2. Show that the vectors y and v are equal in length. The v vector is a sum of two scalar products. I don't see how I can calculate the length directly. Thanks 
May 3rd, 2018, 07:28 AM  #2  
Senior Member Joined: Sep 2015 From: USA Posts: 2,099 Thanks: 1093  Quote:
$y_1,~y_2$ are elements of a vector, not vectors themselves. In this case the easiest route is probably $\begin {align*} &\v\ = \sqrt{v\cdot v} = \\ \\ &\sqrt{(y_1 u_1 + y_2 u_2)\cdot (y_1 u_1 + y_2 u_2)} = \\ \\ &\sqrt{2y_1 y_2 u_1 \cdot u_2 + y_1^2 u_1\cdot u_1 + y_2^2 u_2 \cdot u_2} = \\ \\ &\sqrt{0 + y_1^2 + y_2^2} = \\ \\ &\sqrt{y_1^2 + y_2^2} = \\ \\ &\y\ \end{align*}$  

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