3. The dot product

The dot product or scalar product or inner product of vectors v and w (say in R$^3$) is $v_1 w_1 + v_2 w_2 + v_3 w_3$, denoted v$\cdot$   w. In geometrical terms, v$\cdot$   w$= \vert$v$\vert \vert$w$\vert \cos \theta$, where $\theta$ is the angle between v and w. The dot product makes sense in R$^n$ for any $n$.

The dot product has many uses:

(1)

$\vert$v$\vert = ($v$\cdot$   v$)^{\frac 12}$.

(2)

v$\perp$   w precisely when v$\cdot$   w$= 0$. (The zero vector 0 is considered to be perpendicular to all vectors.)

(3)

The angle $\theta$ between two nonzero vectors v and w in R$^n$ satisfies $\cos \theta = \frac {\mbox{\bf v} \cdot \mbox{\bf w}} {\vert\mbox{\bf v}\vert \vert\mbox{\bf w}\vert}$. Notice, though, that $\cos \theta = \cos (- \theta)$, so the sign of $\theta$ is not determined by the dot product, even in R$^2$, where it is natural to say a counterclockwise angle is positive and a clockwise angle is negative.

(4)

By (), if v$\cdot$   w$> 0$ then the angle between the vectors is between $0 ^\circ$ and $90 ^\circ$ in absolute value; if v$\cdot$   w$<0$, the angle is between $90 ^\circ$ and $180 ^\circ$ in absolute value.

(5)

i$\cdot$   v$,$   j$\cdot$   v$,$   k$\cdot$   v are the components of v, so v$= ($i$\cdot$   v$,$   j$\cdot$   v$,$   k$\cdot$   v$)$.

(6)

The dot product of two unit vectors is simply the cosine of the angle between them.

(7)

By () and () together, you can see that the components of a unit vector u are the cosines of the angles that u makes with i$,$   j$,$   k, or in other words, the cosines of the angles that u makes with the $x$-, $y$-, and $z$-axes. For this reason the components of a unit vector are called direction cosines.

(8)

In a matrix product $AB$, each entry of $AB$ is the dot product of a row of $A$ with a column of $B$.

(9)

Let v, w be vectors in R$^n$, written as row vectors. Then w$^t$ is a column vector. The matrix product v   w$^t$ makes sense and its value is a $1 \times 1$ matrix, which is the same thing as a scalar. By (), this scalar is just v$\cdot$   w. To summarize: v$\cdot$   w$=$   v   w$^t$.

(10)

In R$^3$, the equation of the plane through the point $P_0 = (x_0, y_0, z_{0})$ with normal N$= (a,b,c)$ is N$\cdot ($x$- P_{0}) = 0$, or equivalently,

$a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0$, or equivalently,

$ax + b y + cz + d = 0$, where $d = -$   N$\cdot$   P$_0$.

Similarly, in R$^2$, the equation of the line through the point $P_0 = (x_0, y_{0})$ perpendicular to the vector N$= (a,b)$ is $a(x - x_{0}) + b(y - y_{0}) = 0$. However, usually we deal with lines parametrically, as in §.

Note: For clarity, in this course let's try to use n when we mean a unit normal and N when we mean a normal of any nonzero length.