A vector is a quantity that has both a magnitude and a direction. Vectors can be visualized as arrows. The following figure shows what we mean by the components of a vector
The sum of two vectors has the components
- Explain the addition of vectors in terms of arrows.
The dot product of two vectors is the number
Its importance arises from the fact that it is invariant under rotations. To see this, we calculate
According to Pythagoras, the magnitude of is If we use a different coordinate system, the components of will be different: But if the new system of axes differs only by a rotation and/or translation of the axes, the magnitude of will remain the same:
The squared magnitudes and are invariant under rotations, and so, therefore, is the product
- Show that the dot product is also invariant under translations.
Since by a scalar we mean a number that is invariant under certain transformations (in this case rotations and/or translations of the coordinate axes), the dot product is also known as (a) scalar product. Let us prove that
where is the angle between and To do so, we pick a coordinate system in which In this coordinate system with Since is a scalar, and since scalars are invariant under rotations and translations, the result (which makes no reference to any particular frame) holds in all frames that are rotated and/or translated relative to
We now introduce the unit vectors whose directions are defined by the coordinate axes. They are said to form an orthonormal basis. Ortho because they are mutually orthogonal:
Normal because they are unit vectors:
And basis because every vector can be written as a linear combination of these three vectors — that is, a sum in which each basis vector appears once, multiplied by the corresponding component of (which may be 0):
It is readily seen that which is why we have that
Another definition that is useful (albeit only in a 3-dimensional space) is the cross product of two vectors:
- Show that the cross product is antisymmetric:
As a consequence,
- Show that
Thus is perpendicular to both and
- Show that the magnitude of equals where is the angle between and Hint: use a coordinate system in which and
Since is also the area of the parallelogram spanned by and we can think of as a vector of magnitude perpendicular to Since the cross product yields a vector, it is also known as vector product.
(We save ourselves the trouble of showing that the cross product is invariant under translations and rotations of the coordinate axes, as is required of a vector. Let us however note in passing that if and are polar vectors, then is an axial vector. Under a reflection (for instance, the inversion of a coordinate axis) an ordinary (or polar) vector is invariant, whereas an axial vector changes its sign.)
Here is a useful relation involving both scalar and vector products: