Differential Equations and Linear Algebra, 4th Edition

4.6 Orthogonal Vectors in Rⁿ

In this section we show that the geometrical concepts of distance and angle in n-dimensional space can be based on the definition of the dot product of two vectors in ${\mathbf{\text{R}}}^n$. Recall from elementary calculus that the dot product $\mathbf{\text{u}}\cdot \mathbf{\text{v}}$ of two 3-dimensional vectors $\mathbf{\text{u}}=(u_1,u_2,u_3)$ and $\mathbf{\text{v}}=(v_1,v_2,v_3)$ is (by definition) the sum

$$\mathbf{\text{u}}\cdot \mathbf{\text{v}}=u_1{v}_1+u_2{v}_2+u_3{v}_3$$

of the products of corresponding scalar components of the two vectors.

Similarly, the dot product $\mathbf{\text{u}}\cdot \mathbf{\text{v}}$ of two n-dimensional vectors $\mathbf{\text{u}}=(u_1,u_2,\dots ,u_n)$ and $\mathbf{\text{v}}=(v_1,v_2,\dots ,v_n)$ is defined by

$$\mathbf{\text{u}}\cdot \mathbf{\text{v}}=u_1{v}_1+u_2{v}_2+\cdots \text{+}{u}_n{v}_n$$ (1)

(with one additional scalar product term for each additional dimension). And just as in ${\mathbf{\text{R}}}^3$, it follows readily from the formula in (1) that if u, v, and w are vectors in ${\mathbf{\text{R}}}^n$ and ...