Self Vector Multiplication

Intro

A vector’s dot product with itself equals the square of its Euclidean length, forming the basis for distances, angles, and projections.

Self Vector Multiplication

• Let a vector be expressed in the standard basis $\mathbf{i},\mathbf{j},\mathbf{k}$

$$\mathbf{r}=r_0\mathbf{i}+r_1\mathbf{j}+r_2\mathbf{k}$$

• Take the dot product of the vector with itself:

$$\mathbf{r}\cdot \mathbf{r}=(r_0\mathbf{i}+r_1\mathbf{j}+r_2\mathbf{k})\cdot (r_0\mathbf{i}+r_1\mathbf{j}+r_2\mathbf{k})$$

• Expand distributively:

$$=r_0^2(\mathbf{i}\cdot \mathbf{i})+r_0{r}_1(\mathbf{i}\cdot \mathbf{j})+r_0{r}_2(\mathbf{i}\cdot \mathbf{k})+r_1{r}_0(\mathbf{j}\cdot \mathbf{i})+r_1^2(\mathbf{j}\cdot \mathbf{j})+r_1{r}_2(\mathbf{j}\cdot \mathbf{k})+r_2{r}_0(\mathbf{k}\cdot \mathbf{i})+r_2{r}_1(\mathbf{k}\cdot \mathbf{j})+r_2^2(\mathbf{k}\cdot \mathbf{k})$$

• In Euclidean space, the standard basis vectors are orthonormal, so cross-basis dot products vanish

$$\mathbf{i}\cdot \mathbf{i}=\mathbf{j}\cdot \mathbf{j}=\mathbf{k}\cdot \mathbf{k}=1$$ $$\mathbf{i}\cdot \mathbf{j}=\mathbf{i}\cdot \mathbf{k}=\mathbf{j}\cdot \mathbf{k}=0$$

• Only the matching basis components remain:

$$\mathbf{r}\cdot \mathbf{r}=r_0^2(\mathbf{i}\cdot \mathbf{i})+r_1^2(\mathbf{j}\cdot \mathbf{j})+r_2^2(\mathbf{k}\cdot \mathbf{k})$$

• Since the diagonal terms equal 1:

$$\mathbf{r}\cdot \mathbf{r}=r_0^2+r_1^2+r_2^2$$

• But we know the Vector Magnitude of $r$ is

$$\Vert \mathbf{r}\Vert =sqrt(r_0^2+r_1^2+r_2^2)$$

• So

$$r\cdot r=\Vert \mathbf{r}{\Vert }^2=r_0^2+r_1^2+r_2^2$$