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

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‖ * cosø**which 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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