It is “by definition”. 2 non-zero vectors are stated to be orthogonal when (if and only if) their period product is zero.

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Ok. However why go we define the orthogonality this way?


The algebraic definition

Yet, over there is likewise a geometric definition that the dot product:

a b = ‖a‖ * ‖b‖ * cosøwhich is multiply the length of the an initial vector v the size of the 2nd vector through the cosine that the angle between the two vectors.

And the angle between the two perpendicular vectors is 90°.

When us substitute ø v 90° (cos 90°=0), `a•b` i do not care zero.

I want more than simply a definition. Display me miscellaneous real?

Ok… then let me display you exactly how those two meanings (geometric and also algebraic) agree v each other through the Pythagorean theorem.

Below is a simple recap of the vector norm. They will certainly be supplied in widening Pythagoras theorem come n-dimensions.


The size of n-dimensional vectors

Now, think about two vectors <-1, 2> and also <4, 2>.

Algebraically, <-1, 2> • <4, 2> = -4 + 4 = 0.

Applying the length formula ② to the Pythagorean theorem:



We can expand Pythagorean theorem to n-space.

Geometrically, you can see they room perpendicular together well.


Orthogonality displayed algebraically and geometrically.

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The geometric an interpretation matches through the algebraic definition!

A pop Quiz:

Orthogonal vectors are linearly independent. This sound obvious- yet can you prove it mathematically?

Hint: You can use two definitions. 1) The algebraic definition of orthogonality2) The meaning of direct Independence: The vectors V1, V2, … , Vn room linearly live independence if the equation a1 * V1 + a2 * V2 + … + one * Vn = 0 can only be satisfied through ai = 0 for every i.

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