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 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=[] 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
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 Originally Posted by deltaX 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=[] 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
$v$ is the sum of two weighted vectors.

$y_1,~y_2$ 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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