A *vector* is an arrow with a length and a direction. Just like positions, vectors exist before we measure or describe them. Unlike positions, vectors can mean many different things, such as position vectors, velocities, etc. Vectors are not anchored to particular positions in space, so we can slide a vector around and locate it at any position.

Change:

Two vectors, which may or may not be the same vector. Moving a vector around does not change it: it is still the same vector.

Some textbooks differentiate between *free
vectors*, which are free to slide around, and
*bound vectors*, which are anchored in space. We
will only use free vectors.

We will use the over-arrow notation \( \vec{a} \) for vector quantities. Other common notations include bold \( \boldsymbol{a} \) and under-bars \( \underline{a} \). For unit (length one) vectors we will use an over-hat \( \hat{a} \).

While a vector represents magnitude and direction, a *scalar* is a number that represents a magnitude, but with no directional information. Some examples of scalar quantities can be mass, length, time, speed, or temperature.

Vectors can be added or subtracted together, using the parallelogram law of addition or the head-to-tail rule.

A *unit vector* is any vector with a length of
one. We use the special over-hat notation \( \hat{a} \) to
indicate when a vector is a unit vector. Any non-zero vector
\( \vec{a} \) gives a unit vector \( \hat{a} \) that specifies the
direction of \( \vec{a} \).

Normalization to unit vector. #rvv-eu

Any vector can be written as the product of its length and direction:

Vector decomposition into length and direction. #rvv-ei

Three vectors and their decompositions into lengths and directional unit vectors.

Vectors can be written as a magnitude (length) multiplied by the unit vector in the same direction as the original vector.

The length of a vector \( \vec{a} \) is written
either \( \| \vec{a} \| \) or just plain \(a\). The
length can be computed using *Pythagorus’
theorem*:

Pythagorus' length formula. #rvv-ey

Warning: Length must be computed in a single basis. #rvv-wl

Computing the length of a vector using Pythagorus' theorem.

Some common integer vector lengths are \( \vec{a} = 4\hat\imath + 3\hat\jmath \) (length \(a = 5\)) and \( \vec{b} = 12\hat\imath + 5\hat\jmath \) (length \(b = 13\)).

Warning: Adding vectors does not add lengths. #rvv-wa

The direction of a vector can be written as a unit vector by dividing the vector components by the vector magnitude.

Alternatively, the vector components can be determined geometrically via the angles of each component with respect to the Cartesian Axes.

The *dot product* (also called the *inner
product* or *scalar product*) is defined by

Dot product from components.

An alternative expression for the dot product can be given in terms of the lengths of the vectors and the angle between them:

Dot product from length/angle. #rvv-ed

The fact that we can write the dot product in terms of components as well as in terms of lengths and angle is very helpful for calculating the length and angles of vectors from the component representations.

Length and angle from dot product. #rvv-el

If two vectors have zero dot product \( \vec{a} \cdot \vec{b} = 0 \) then they have an angle of \( \theta = 90^\circ = \frac{\pi}{2}\rm\ rad \) between them and we say that the
vectors are *perpendicular*, *orthogonal*, or
*normal* to each other.

In 2D we can easily find a perpendicular vector by rotating \( \vec{a} \) counterclockwise with the following equation.

Counterclockwise perpendicular vector in 2D. #rvv-en

In 2D there are two perpendicular directions to a given vector \( \vec{a} \), given by \( \vec{a}^\perp \) and \( -\vec{a}^\perp \). In 3D there is are many perpendicular vectors, and there is no simple formula like #rvv-en for 3D.

The perpendicular vector \( \vec{a}^\perp \) is always a \( +90^\circ \) rotation of \( \vec{a} \).

Dot product symmetry. #rvi-ed

Dot product vector length. #rvi-eg

Dot product bi-linearity. #rvi-ei

The cross product can be defined in terms of components by:

Cross product in components. #rvv-ex

It is sometimes more convenient to work with cross products of individual basis vectors, which are related as follows.

Cross products of basis vectors. #rvv-eo

Warning: The cross product is not associative. #rvv-wc

Rather than using components, the cross product can be defined by specifying the length and direction of the resulting vector. The direction of \( \vec{a} \times \vec{b} \) is orthogonal to both \( \vec{a} \) and \( \vec{b} \), with the direction given by the right-hand rule. The magnitude of the cross product is given by:

Cross product length. #rvv-el2

This second form of the cross product definition can also be related to the area of a parallelogram.

A useful special case of the cross product occurs when vector \( \vec{a} \) is in the 2D \( \hat\imath,\hat\jmath \) plane and the other vector is in the orthogonal \( \hat{k} \) direction. In this case the cross product rotates \( \vec{a} \) by \( 90^\circ \) counterclockwise to give the perpendicular vector \( \vec{a}^\perp \), as follows.

Cross product of out-of-plane vector \( \hat{k} \) with 2D vector \( \vec{a} = a_1\,\hat\imath + a_2\,\hat\jmath \). #rvv-e9

Cross product anti-symmetry. #rvi-ea

Cross product self-annihilation. #rvi-ez

Cross product bi-linearity. #rvi-eb2

The cross products of 3D vectors can be calculated by taking the determinant of specific components of the two vectors. It's best to start by writing a 3x3 matrix with the \( \hat{\imath} \), \( \hat{\jmath} \), and \( \hat{k} \) vectors in the first row and the two vectors you are taking the cross product of in the next two rows. See the example below of the cross product between \( \vec{A} \) and \( \vec{B} \):

The projection and complementary projection are:

Projection of \(\vec{a}\) onto \(\vec{b}\). #rvv-ep

Complementary projection of \(\vec{a}\) with respect to \(\vec{b}\). #rvv-em

Adding the projection and the complementary projection of a vector just give the same vector again, as we can see on the figure below.

Projection of \( \vec{a} \) onto \( \vec{b} \) and the complementary projection.

As we see in the diagram above, the complementary projection is orthogonal to the reference vector:

Complementary projection is orthogonal to the reference. #rvv-er