$Consider the vector v = (3, 4, 5)^T, calculate the orthogonal projection$

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$Consider the vector v = (3, 4, 5)^T, calculate the orthogonal projection$

[Distance from a vector to a plane]. Consider the vector v = (3, 4, 5)^T and the plane P: x + y + z = 0, which is a subspace of R^3 and has dimension 2

(a) Calculate the orthogonal projection of V over P
(b) Calculate the orthogonal projection of V over n, the normal vector to P

Linear Algebra

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The normal vector to the plane is $n=(1,1,1)$.
(b) Orthogonal projection of $V$ onto $n=(1,1)$ is
\[\frac{V\cdot n}{n \cdot n}n=\frac{(3,4,5)\cdot(1,1,1)}{(1,1,1)\cdot (1,1,1)}(1,1,1)=\frac{3+4+5}{1+1+1}(1,1,1)\]
\[=\frac{12}{3}(1,1,1)=(4,4,4).\]
(a) Orthogonal projection of $V$ onto the plane is
\[(3,4,5)-(4,4,4)=(-1,0,1).\]

See this for more details: https://www.maplesoft.com/support/help/maple/view.aspx?path=MathApps%2FProjectionOfVectorOntoPlane

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$Consider the vector v = (3, 4, 5)^T, calculate the orthogonal projection$

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