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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*} Tags length, vectors Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mms Math 10 July 31st, 2015 07:11 AM joshbeldon Calculus 3 February 26th, 2015 11:58 AM iqbad Algebra 1 April 12th, 2013 03:38 PM summychan Calculus 14 October 8th, 2012 08:15 PM amin7905 Algebra 4 July 19th, 2012 01:08 PM

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