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The cross product of vector A with the zero vector, yielding the zero vector, since the zero vector does not have a direction or magnitude to contribute to the cross product.

The difference between vectors A and B, resulting in a vector pointing from B towards A.

The dot product of vectors A and B, resulting in a scalar representing the magnitude of A multiplied by the magnitude of B and the cosine of the angle between them.

The subtraction of vector A from itself, yielding the zero vector.

The cross product of vector A with the scalar multiplication of vector B by k, resulting in a vector that is k times larger than $\mathbf{A} \times \mathbf{B}$, keeping the same direction given by the right-hand rule if k is positive.

The distributive property of the dot product, yielding the same result as $\mathbf{A} \cdot \mathbf{B} - \mathbf{A} \cdot \mathbf{C}$. The dot product is distributed over the subtraction of vectors B and C.

The projection of vector B onto vector A, resulting in a vector that lies on A and has a length equal to the component of B that is parallel to A.

The scalar multiplication of vector A by a scalar k, where the result is a vector k times as long in the same or opposite direction, depending on the sign of k.

The sum of the vectors A and B, resulting in a vector that represents the combined magnitude and direction of A and B.

The distributive property of the dot product, yielding the same result as $\mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$. The dot product is distributed over the addition of vectors B and C.

The dot product of the zero vector with vector A, yielding a scalar of 0 since the zero vector has no magnitude.

Scalar multiplication distributed over vector subtraction, resulting in $k\mathbf{A} - k\mathbf{B}$. The difference of the vectors is multiplied by scalar k.

The dot product of vector A with the scalar multiplication of vector B by k, which can be factored out to result in $k(\mathbf{A} \cdot \mathbf{B})$.

The negation of vector A, resulting in a vector with the same magnitude as A but pointing in the exact opposite direction.

Scalar multiplication distributed over vector addition, resulting in $k\mathbf{A} + k\mathbf{B}$. The sum of the vectors is multiplied by scalar k.

The division of vector A by a non-zero scalar k, resulting in a vector each of whose components is divided by k.

The cross product of vector A with the negation of vector B, yielding a vector that is opposite in direction to $\mathbf{A} \times \mathbf{B}$ and has the same magnitude.

The magnitude or norm of vector A, which is a scalar representing the length of A.

The cross product of vectors A and B, resulting in a vector that is perpendicular to both A and B and has a magnitude equal to the area of the parallelogram that A and B span.