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Scalars and vectors

Scalars and vectors

A scalar $s$ is a type of physical quantity that has a magnitude (size) but no direction.

Length, charge, temperature, energy, speed and distance are scalar quantities.

A vector $\vecphy{v}$ is a type of physical quantity that has both magnitude and direction.

Displacement, velocity, acceleration and force are vector quantities.

A vector has a magnitude $\vert \vecphy{v}\vert$ or $v$, which is a scalar and corresponds to the length (absolute value or modulus) of the vector.

A vector quantity is represented graphically as an arrow pointing in the direction of the quantity.

Vectors have direction: $u$ and $v$ have the same direction, and they are both opposite in direction to $w$

When vectors are added to or subtracted from each other, the directions of each of the vectors have to be taken into account.

The magnitude of the resultant vector is generally not equivalent to the sum or difference of the magnitudes of individual vectors

Vectors can be added graphically by moving one of the vectors (A) so that its tail joins the head of the other vector (B). The resultant vector is the arrow between the tail of B and the head of A.

A vector subtracted from another follows the same rules, except that the vector being subtracted has its direction reversed. The resultant vector of $\vecphy{A}-\vecphy{B}$ is drawn from the starting point of A to the ending point of B projected in the opposite direction.

A vector $\vecphy{v}$ has two perpendicular components, usually along the horizontal $(x)$ and vertical $(y)$ axis. This is useful when the vector points in the direction of, say, an inclined plane. These components are given by:\begin{align*}v_{y}=v\sin\theta\\v_{x}=v\cos\theta\end{align*}$\theta$ is the angle between the vector and the line taken to be the horizontal. Note that $\vecphy{v}=\vecphy{v}_{y}+\vecphy{v}_{x}$ as defined. The magnitude of the vector $\vecphy{v}$ is given by: $\vert \vecphy{v}\vert=\sqrt{v_{x}^{2}+v_{y}^{2}}$ (by Pythagoras' theorem).
We say that we have projected the vector $\vecphy{v}$ along the horizontal $(x)$ and vertical $(y)$ axis.