angle between two vector when one vector is zero

Question

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 :

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

I cant figure out the angle between these two vectors . Please help me .

Answer 1

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.

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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).