Multiplication of a vector $\mathbf {b} \,$ by a scalar $\lambda \,$ has the effect of stretching or shrinking the vector.

You can form a unit vector ${\hat {\mathbf {b} }}\,$ that is parallel to $\mathbf {b} \,$ by dividing by the length of the vector $|\mathbf {b} |\,$. Thus,

where $\theta \,$ is the angle between the two vectors (see Figure 2(b)).

If $\mathbf {a} \,$ and $\mathbf {b} \,$ are perpendicular to each other, $\theta =\pi /2\,$ and $\cos(\theta )=0\,$. Therefore, ${\mathbf {a} }\cdot {\mathbf {b} }=0$.

The dot product therefore has the geometric interpretation as the length of the projection of $\mathbf {a} \,$ onto the unit vector ${\hat {\mathbf {b} }}\,$ when the two vectors are placed so that they start from the same point (tail-to-tail).

The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as

where $\theta \,$ is the angle between $\mathbf {a} \,$ and $\mathbf {b} \,$, and ${\hat {\mathbf {c} }}\,$ is a unit vector perpendicular to the plane containing $\mathbf {a} \,$ and $\mathbf {b} \,$ in the right-handed sense.

In terms of the orthonormal basis $(\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})\,$, the cross product can be written in the form of a determinant