Let yrightarrow 4 7 6 and urightarrow 1 6 6 Write yrightarr

Let y^rightarrow [-4 -7 6] and u^rightarrow = [1 -6 6]. Write y^rightarrow as the sum of two orthogonal vectors, x^rightarrow_1 in Span {u} and x^rightarrow_2 orthogonal to u^rightarrow.

So, we want : y = x1+ x2

x1 = ku ; x1*x2 =0 ; x1 = ( k , -6k , 6k)

( 4 , -7 , 6) = x1 +x2

Now take dot product with x1 on both sides:

(4, -7 , 6)(k , -6k , 6k) = 73k^2 +0

k can be cancelled out : ( 4 +42+36) = 73k

k = 82/73

So, x1 = ku = [ 82/73 , -492/73 , 216/73 ] ^T

x2 = y - x1 = [ -4, -7 , 6]^T - [ 82/73 , -492/73 , 216/73 ] ^T

= [ -374/73 , -19/73 , -222/73]^T

$Let y^rightarrow [-4 -7 6] and u^rightarrow = [1 -6 6]. Write y^rightarrow as the sum of two orthogonal vectors, x^rightarrow_1 in Span {u} and x^rightarrow_2$

Let yrightarrow 4 7 6 and urightarrow 1 6 6 Write yrightarr

Solution

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