You are watching: If the dot product of two nonzero vectors v1 and v2 is zero what does this tell us
Ok. But why did we define the orthogonality this way?
The algebraic definition
Yet, over there is also a geometric definition the the period product:
a • b = ‖a‖ * ‖b‖ * cosøwhich is multiplying the size of the an initial vector through the length of the second vector v the cosine that the edge between the two vectors.
And the angle between the 2 perpendicular vectors is 90°.
When us substitute ø with 90° (cos 90°=0), `a•b` i do not care zero.
I want much more than simply a definition. Show me miscellaneous real?
Ok… then let me show you how those two meanings (geometric and also algebraic) agree v each other through the Pythagorean theorem.
Below is a simple recap the the vector norm. They will certainly be offered in expanding Pythagoras theorem come n-dimensions.
The size of n-dimensional vectors
Now, take into consideration two vectors <-1, 2> and <4, 2>.
Algebraically, <-1, 2> • <4, 2> = -4 + 4 = 0.
Applying the size formula ② come the Pythagorean theorem:
We can extend Pythagorean theorem to n-space.
Geometrically, you can see they are perpendicular together well.
Orthogonality displayed algebraically and also geometrically.
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The geometric an interpretation matches with the algebraic definition!
A pop Quiz:
Orthogonal vectors room linearly independent. This sound obvious- yet can friend prove that mathematically?
Hint: You deserve to use 2 definitions. 1) The algebraic definition of orthogonality2) The meaning of linear Independence: The vectors V1, V2, … , Vn space linearly independent if the equation a1 * V1 + a2 * V2 + … + an * Vn = 0 can only it is in satisfied by ai = 0 for every i.
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