Euclidean Vector Multiplication

Intro

In Euclidean space, the multiplication of two vectors corresponds to the inner product (dot product). Because the Euclidean basis is orthonormal, vector multiplication only produces non-zero terms when the basis vectors match:

• $\mathbf{i}\cdot \mathbf{i}$
• $\mathbf{j}\cdot \mathbf{j}$
• $\mathbf{k}\cdot \mathbf{k}$

Euclidean Vector Multiplication

• Let two vectors (standard basis $\mathbf{i},\mathbf{j},\mathbf{k}$)

$$\mathbf{r}=r_1\mathbf{i}+r_2\mathbf{j}+r_3\mathbf{k}$$ $$\mathbf{s}=s_1\mathbf{i}+s_2\mathbf{j}+s_3\mathbf{k}$$

• Multiply distributively:

$$\mathbf{r}\cdot \mathbf{s}=(r_1\mathbf{i}+r_2\mathbf{j}+r_3\mathbf{k})(s_1\mathbf{i}+s_2\mathbf{j}+s_3\mathbf{k})$$ $$=r_1{s}_1\mathbf{ii}+r_1{s}_2\mathbf{ij}+r_1{s}_3\mathbf{ik}+r_2{s}_1\mathbf{ji}+r_2{s}_2\mathbf{jj}+r_2{s}_3\mathbf{jk}+r_3{s}_1\mathbf{ki}+r_3{s}_2\mathbf{kj}+r_3{s}_3\mathbf{kk}$$

• 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:

$$\mathbf{r}\cdot \mathbf{s}=r_1{s}_1(\mathbf{i}\cdot \mathbf{i})+r_2{s}_2(\mathbf{j}\cdot \mathbf{j})+r_3{s}_3(\mathbf{k}\cdot \mathbf{k})$$

• Since the diagonal terms equal 1:

$$\mathbf{r}\cdot \mathbf{s}=r_1{s}_1+r_2{s}_2+r_3{s}_3$$

Related

• Self Vector Multiplication