The Dot Product and Orthogonality

Definition The dot product of $\mathbf{u} = \langle u_1, u_2 \rangle$ and $\mathbf{v} = \langle v_1, v_2 \rangle$ is the real number

For vectors in space it is $u_1 v_1 + u_2 v_2 + u_3 v_3$. This sum of products is the component formula for the dot product. The result is a scalar, not a vector, which is why some books call it the scalar product.

Definition For all vectors $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ of the same dimension and every scalar $c$

and the dot product of a vector with itself is its squared length,

Also $\mathbf{0} \cdot \mathbf{v} = 0$ for every $\mathbf{v}$. Each rule follows from the component formula and the rules for real numbers.

Definition Let $\mathbf{u}$ and $\mathbf{v}$ be nonzero vectors, drawn from a common initial point. The angle between $\mathbf{u}$ and $\mathbf{v}$ is the angle $\theta$ between these two arrows with

It does not depend on the order of the vectors and has no sign. The angle is $0$ when the vectors point the same way and $\pi$ when they point in opposite ways.

Definition If $\mathbf{u}$ and $\mathbf{v}$ are nonzero vectors and $\theta$ is the angle between them, then

Since $\theta$ lies in $[0, \pi]$, which is the range of the inverse cosine, the angle itself is

To see why the formula holds, draw $\mathbf{u}$ and $\mathbf{v}$ from one point. If they do not lie along one line, they form a triangle whose third side is $\mathbf{u} - \mathbf{v}$ and whose angle opposite that side is $\theta$. Expanding $(\mathbf{u} - \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v})$ with the rules above, and applying the law of cosines to the same side, gives

Canceling the squared lengths and dividing by $-2$ leaves the formula. When the vectors lie along one line there is no triangle, and the formula can be checked directly with $\cos \theta = \pm 1$.

Definition For nonzero $\mathbf{u}$ and $\mathbf{v}$ the lengths in the geometric formula are positive, so $\mathbf{u} \cdot \mathbf{v}$ has the sign of $\cos \theta$. It is positive when $0 \le \theta < \pi/2$, zero when $\theta = \pi/2$, and negative when $\pi/2 < \theta \le \pi$. The factor $\lVert \mathbf{v} \rVert \cos \theta$ is the signed length of the shadow that $\mathbf{v}$ casts on the line through $\mathbf{u}$, counted positive when the shadow points the way $\mathbf{u}$ points, so

The next lesson calls this shadow length the scalar component.

Definition Two vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal when

For nonzero vectors this happens exactly when the angle between them is $\pi/2$, so for them orthogonal means perpendicular. By this definition the zero vector is orthogonal to every vector.

Definition Two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel when $\mathbf{v} = c\mathbf{u}$ for some scalar $c$. Then $c \ne 0$, and the angle between them is $0$ when $c > 0$ and $\pi$ when $c < 0$. Because $\lvert \cos \theta \rvert \le 1$, the geometric formula shows that nonzero vectors satisfy

and equality holds exactly when $\cos \theta = \pm 1$, that is, exactly when the vectors are parallel.

In the plane, nonzero $\langle u_1, u_2 \rangle$ and $\langle v_1, v_2 \rangle$ are parallel exactly when $u_1 v_2 - u_2 v_1 = 0$.

Definition The zero vector has length $0$ and no direction. So the angle between $\mathbf{0}$ and another vector is not defined, and the formula for $\cos \theta$ cannot be used, since its denominator $\lVert \mathbf{u} \rVert \, \lVert \mathbf{v} \rVert$ is $0$. Because $\mathbf{0} \cdot \mathbf{v} = 0$, the zero vector counts as orthogonal to every vector. Books differ on whether it also counts as parallel to every vector, since $\mathbf{0} = 0\mathbf{v}$, which is why statements about angles and parallel vectors are made for nonzero vectors. Two more cautions follow. A dot product of $0$ does not mean that one of the vectors is $\mathbf{0}$, and $\mathbf{u} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{w}$ does not mean $\mathbf{v} = \mathbf{w}$.