A physics quantity of type vector must be specified with a magnitude and a direction [1], or vector is simply a quantity that has both magnitude and direction [2], where the magnitude is sometimes also referred to as size or length [3]. Further reading will explain vector as element of vector space [4, 5]. Here we will require only the simple definition of a vector. Vector can be drawn using a directed line segment [6]. Two examples about vector, which differs it from scalar, are displacement compared to distance and velocity compared to speed [7].
A vector $\vec{r}$ in a 2-d (two-dimension) space can be written as
with $r_x \equiv x$ and $r_y \equiv y$ are vector components in $x$ and $y$ directions, respectively. Each direction is indicated by each unit vector, i.e. $\hat{x} \equiv \hat{i}$, $\hat{y} \equiv \hat{j}$. In figures a vector, or in general a matrix, can also represent in the form of
with examples in Fig. 1. A vector can be illustrated graphically using vector diagram in the form of an arrow [8], where length of the arrow represent magnitude of the vector and direction of the arrow is direction of the vector.
svg 150 100 #fafafa fig:vec-arrow-1|Three arrows a, b, c representing three different vectors.
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There are three arrows shown in Fig. 1, which are $\mathbf{a} \equiv \vec{a}$, $\mathbf{b} \equiv \vec{b}$, and $\mathbf{c} \equiv \vec{c}$, as previously described in Eqn. \eqref{eqn:vec-4}. Suppose that there is a vector $\vec{c} = 4 \hat{i} + 3 \hat{j}$ as shown in Fig. 2.
svg 240 200 #fafafa fig:vec-arrow-2|A vector $\mathbf{c} \equiv \vec{c} = 4 \hat{i} + 3 \hat{j}$ in $xy$ plane.
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The value of 4 is projection of vector $\vec{c}$ along $x$ direction and 3 along $y$ direction as shown with darker dashed line in Fig. 2. Any vector can also be drawn like $\vec{c}$. The vector $\vec{c}$ can be drawn from any point, not necessary from (0, 0), as long the length and direction are the same.
Magnitude of a vector in Eqn. \eqref{eqn:vec-2} can be obtained using
\begin{equation}
\label{eqn:vec-2-magnitude}
r = |\vec{r}| = \sqrt{x^2 + y^2},
\end{equation}
which can also be produced using square root of dot product of the vector of itself. Dot product is one of multiplication operation of two vectors. The use of Eqn. \eqref{eqn:vec-2-magnitude} for vector in Fig. 2 will produce $c = |\vec{c}| = 5$.
exer
Explain about a vector using its two important features.
Tell which ones are scalar and which ones are vector from following physical quantities: speed, dispacement, velocity, distance, force, pressure, mass, density, torque, momentum.
Explain what direction the each unit vector $\hat{i}$ and $\hat{j}$ represents. See Eqns. \eqref{eqn:vec-1} - \eqref{eqn:vec-3} when necessary.
Draw a vector $\vec{c} = \hat{i} - 2 \hat{j}$ using previous example in Fig. 2.
Eqn. \eqref{eqn:vec-2-magnitude} is magnitude for vector in Eqn. \eqref{eqn:vec-2}, write the magnitude for for vector in Eqn. \eqref{eqn:vec-1}.