Scalars and Vectors

Scalars, such as temperature, have magnitude only. They are specified by a number with a unit (10°C) and obey the rules of arithmetic and ordinary algebra. Vectors, such as displacement, have both magnitude and direction (5 m, north) and obey the rules of vector algebra.

Adding Vectors Geometrically

Two vectors $\vec a$ and $\vec b$ may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum $\vec s$. To subtract $\vec b$ from $\vec a$, reverse the direction of $\vec b$ to get $-\vec b$; then add $-\vec b$ to $\vec a$. Vector addition is commutative

and obeys the associative law.

Components of a Vector

The (scalar) components $a_x$ and $a_y$ of any two-dimensional vector along the coordinate axes are found by dropping perpendicular lines from the ends of onto the coordinate axes. The components are given by

where $\theta$ is the angle between the positive direction of the $x$ axis and the direction of $\vec a$. The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation (direction) of the vector by using

Unit Vectors

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point, that is, to specify a direction. The unit vectors in the positive directions of the x, y, and z axes are labeled $\hat i$, $\hat j$, and $\hat k$, where the hat $\hat{}$ is used instead of an overhead arrow as for other vectors.

Essentially, these can be used to give direction by multiplying with numerical magnitudes.

A vector $\vec a$ can be written in terms of unit vectors as

in which $a_x\hat i$, $a_y\hat j$, and $a_z\hat k$ are the vector components of $\vec a$ and $a_x$, $a_y$, and $a_z$ are its scalar components.

Adding Vectors by Components

We can add vectors geometrically on a sketch or directly on a vector-capable calculator. A third way is to combine their components axis by axis.

To add vectors in component form, we use the rules

Here $\vec a$ and $\vec b$ are the vectors being added, and $\vec r$ is the vector sum. After summing each of the separate components of the vector, it can be represented in unit-vector notation or magnitude-angle notation.

Product of a Scalar and a Vector

The product of a scalar $s$ and a vector $\vec v$ is a new vector whose magnitude is $sv$ and whose direction is the same as that of $\vec v$ if $s$ is positive, and opposite that of $\vec v$ if $s$ is negative. To divide $\vec v$ by $s$, multiply $\vec v$ by $\frac{1}{s}$.

The Scalar Product

The scalar (or dot) product of two vectors $\vec a$ and $\vec b$ is written $\vec a \cdot \vec b$ and is the scalar quantity given by $\vec a \cdot \vec b = ab \space cos \space \phi$, in which $\phi$ is the angle between the directions of $\vec a$ and $\vec b$. A scalar product is the product of the magnitude of one vector and the scalar component of the second vector along the direction of the first vector. In unit-vector notation,

which may be expanded according to the distributive law. Note that $\vec a \cdot \vec b = \vec b \cdot \vec a$.

The Vector Product

The vector (or cross) product of two vectors $\vec a$ and $\vec b$ is written $\vec a \times \vec b$ and is a vector $\vec c$ whose magnitude $c$ is given by:

in which $\phi$ is the smaller of the angles between the directions of $\vec a$ and $\vec b$. The direction of $\vec c$ is perpendicular to the plane defined by $\vec a$ and $\vec b$ and is given by a right-hand rule. Note that $\vec a \times \vec b = -(\vec b \times \vec a)$. In unit-vector notation,

which we may expand with the distributive law.