Proof: This follows simply because any set of n linearly independent vectors in Rn is a basis. x. (Note then than x · x = |x|2.) Definition: A basis B = {x1,x2,…,xn} of Rn is said to be an orthogonal basis if the elements of B are pairwise orthogonal, that is xi · xj whenever i = j.

Remember that two vectors a and b are perpendicular to each other if their dot product is zero. Here in your case , say a = (i + 2 j + 4 k ) and b = (2i – j ) then it is easy to see that a . b = ( i + 2 j + 4 k ) . ( 2 i – j ) = 1*2 – 2* 1 = 0 .

We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero. Definition. We say that a set of vectors { v1, v2., vn} are mutually or- thogonal if every pair of vectors is orthogonal.

Now any set of linear independent vectors would be a scalar multiple of these two vectors that form a Basis for R2 hence they have to be orthogonal. …

n. The two-dimensional graphic representation of an object formed by the perpendicular intersections of lines drawn from points on the object to a plane of projection. Also called orthographic projection.

If V0 is a one-dimensional subspace spanned by a vector v then p = (x,v) (v,v) v. If v1,v2,…,vn is an orthogonal basis for V0 then p = (x,v1) (v1,v1) v1 + (x,v2) (v2,v2) v2 + ··· + (x,vn) (vn,vn) vn. (x,vj ) (vj ,vj ) (vj ,vi ) = (x,vi ) (vi ,vi ) (vi ,vi ) = (x,vi ) =⇒ (x−p,vi ) =0 =⇒ x−p ⊥ vi =⇒ x−p ⊥ V0.

A basis B for a subspace of is an orthogonal basis for if and only if B is an orthogonal set. Similarly, a basis B for is an orthonormal basis for if and only if B is an orthonormal set. If B is an orthogonal set of n nonzero vectors in , then B is an orthogonal basis for .

The special thing about an orthonormal basis is that it makes those last two equalities hold. With an orthonormal basis, the coordinate representations have the same lengths as the original vectors, and make the same angles with each other.

Two functions are orthogonal with respect to a weighted inner product if the integral of the product of the two functions and the weight function is identically zero on the chosen interval. Finding a family of orthogonal functions is important in order to identify a basis for a function space.

adj. 1. describing a set of axes at right angles to one another, which in graphical representations of mathematical computations (such as factor analysis) and other research indicates uncorrelated (unrelated) variables. Compare oblique.

A related term, orthogonal projection, describes a method for drawing three-dimensional objects with linear perspective. It refers to perspective lines, drawn diagonally along parallel lines that meet at a so-called “vanishing point.” Such perspective lines are orthogonal, or perpendicular to one another.

In debate(?), “orthogonal” to mean “not relevant” or “unrelated” also comes from the above meaning. If issues X and Y are “orthogonal”, then X has no bearing on Y. If you think of X and Y as vectors, then X has no component in the direction of Y: in other words, it is orthogonal in the mathematical sense.