# angle between two vector when one vector is zero

I am trying to find angle between two vector . I know the theory . If u and v are two vectors then the angle between these two vector is defined as the following theorem :

$$\theta=\arccos\left(\frac{\operatorname{Re}(u\cdot v)}{\|u\|\|v\|}\right)$$

where the inner product u⋅vu⋅v is defined to be

$$u\cdot v=\sum_{k=0}^{n-1} u_k\overline{v_k}$$

But when when one of the two vectors is zero , then what will be the angle between the two vector ? Suppose what will be angle of the following two vector :

$$\vec{u} = 5\hat{i} + 2\hat{j}+3\hat{k}$$ $$\vec{v} = 0\hat{i} + 0\hat{j}+0\hat{k}$$

• Geometrically it doesn't make much sense to consider an angle between a line and a point. In the same fashion it doesn't make any sense to consider the angle between a (non-zero) vector and the zero vector. Using the definition above you immediately run into troubles since $\|v\|=0$, thus you divide by zero. On the other hand, $\left\langle w,v \right\rangle=0$, so you could argue that they are perpendicular. Aug 30, 2017 at 6:14
When one of the two vectors is $0$, the angle between them is not defined.
One way to look at this is that the zero vector doesn't really have a "direction". If a vector $v$ is non-zero, then the direction of that vector can, in some sense, be represented by the vector $\frac{v}{\|v\|}$, and $\frac{0}{\|0\|}$ is not defined. And since the angle between two vectors is really the angle between their directions, it makes sense you can't plug a $0$ vector into the equation.
However, you can still say that the vectors are orthogonal, because their dot product is $0$ - the zero vector, therefore, is orthogonal to every other vector (including itself).