Can a cross product be negative?

Hint: The process of finding a negative valued cross product is to find the angle of the vectors. The angle describes the direction of the product vector which can be considered as the negative sign depending on the positive side.

Complete step by step solution:
Given two linearly independent vectors a and b, the cross product, $a\times b$ is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
The cross product of two vectors is itself a vector, and vectors do not have a meaningful notion of positive or negative.
The angle of the vector defines the direction of the vector which in turn gives the sign of the vector as positive or negative.
The formula for cross product is $\overrightarrow{a}\times \overrightarrow{b}=\left| \overrightarrow{a} \right|\times \left| \overrightarrow{b} \right|\times \sin \alpha $ where the angle between the vectors is $\alpha $.
If we have to answer it with respect to angle then we say that if the angle between two vectors varies between ${{180}^{\circ }} < \alpha < {{360}^{\circ }}$, then cross product becomes negative because for ${{180}^{\circ }} < x < {{360}^{\circ }}$, $\sin \alpha $ is negative.

Note: When we find the cross product of two unit vectors, then the sign of the vector just signifies the direction of the vector. But in the case of a dot product of two unit vectors the resultant vector is cosine, that can be negative or positive.