You are watching: If the dot product of two nonzero vectors is zero
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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