Some Basic Concepts

Definition

A vector is a quantity that has both magnitude and direction.

A directed line segment is a vector.

• It is denoted as $\overrightarrow{AB}$ or simply $\vec{a}$, and read as "vector $AB$" or "vector $a$".
• The point $A$ where the vector starts is called the initial point.
• The point $B$ where it ends is called the terminal point.
• The magnitude (or length) of the vector is the distance between its initial and terminal points, denoted as $|\overrightarrow{AB}|$, $|\vec{a}|$, or $a$.
• The arrow indicates the direction of the vector.

Note: Since length is never negative, the notation $|\vec{a}| < 0$ has no meaning.

Intuition

Think of a vector as an arrow — it tells you how far to go (magnitude) and which way to go (direction). A plain number (like 5 kg) has only magnitude; a vector (like "5 km east") has both.

Example …

Fig 10.1 is a three‑panel diagram that builds the idea of a vector from the most basic geometric object: a straight line.

Panel (i) shows a straight line l drawn as a dotted line. A solid arrow runs upward along it. This arrow gives the line a direction — one of the two possible ways to travel along l. The line together with this chosen direction is called a directed line.

Panel (ii) shows the same dotted line l, but now the solid arrow points downward. This is the opposite direction. The two panels together make the point: any straight line can be given two directions, and a directed line is simply a line with one of those directions prescribed.

Panel (iii) is the crucial step. Instead of the whole infinite line, we now restrict ourselves to the line segment from point $A$ to point $B$. The segment is drawn as a solid arrow from $A$ to $B$, and the vector is labelled $\vec{a}$ (or $\overrightarrow{AB}$). This is a directed line segment — it has a definite length (the distance $AB$) and a definite direction (from $A$ to $B$). The textbook then defines a vector as exactly this: a quantity that has both magnitude and direction.

The figure does not show any axes or coordinates — it is purely geometric. Its purpose is to separate the concept of direction (panels i and ii) from the concept of a vector as a directed segment with a specific length (panel iii). Once this foundation is laid, the textbook moves to the coordinate system and introduces the position vector of a point $P(x,y,z)$ relative to the origin $O$:

with magnitude

From there, the direction cosines are defined. If $\alpha, \beta, \gamma$ are the angles that $\vec{r}$ makes with the positive $x$, $y$, and $z$ axes respectively, then

$\cos\alpha = \frac{x}{r},\quad \cos\beta = \frac{y}{r},\quad \cos\gamma = \frac{z}{r},$ …

Fig 10.2 is the textbook’s first visual bridge between the coordinate geometry you studied in Class XI and the vector language of Class XII. It shows two related 3‑D sketches, both drawn in a right‑handed coordinate system with the Z‑axis pointing straight up, the Y‑axis to the right, and the X‑axis coming toward the lower left. The origin is labelled O.

Panel (i) — the left sketch — introduces the position vector. A point $P(x,y,z)$ is placed somewhere in the first octant (all coordinates positive). A solid arrow, usually coloured indigo in the printed book, runs from O to P. This arrow is the vector $\vec{r} = \overrightarrow{OP}$. Its length is the distance from the origin to P, which by the distance formula is

The dashed lines from P back to the axes (dropping perpendiculars to the coordinate planes) make it clear that $x$, $y$, $z$ are the components of $\vec{r}$ along the three axes. So the figure’s first message is: every point in space corresponds to a unique position vector from the origin, and the coordinates of the point are exactly the components of that vector.

Panel (ii) — the right sketch — generalises the idea. Now three different points $A$, $B$, $C$ are shown, each with its own position vector from O: $\vec{a} = \overrightarrow{OA}$, $\vec{b} = \overrightarrow{OB}$, $\vec{c} = \overrightarrow{OC}$. All three vectors are coinitial — they share the same starting point O. This panel prepares you for the notation used throughout the chapter: the position vector of any point $X$ is written as $\vec{x}$ (or $\overrightarrow{OX}$). It also sets up the visual habit of thinking of vectors as arrows radiating from a common origin, which is essential when you later add vectors or resolve them into components.

The textbook uses this figure to lead directly into the definition of direction cosines (Fig 10.3, which follows immediately). In that next figure, the same position vector $\vec{r} = \overrightarrow{OP}$ is shown with the angles $\alpha$, $\beta$, $\gamma$ that it makes with the positive $X$, $Y$, $Z$ axes. From the right‑angled triangles visible in the sketch — triangle $OAP$ in the $XZ$‑plane, $OBP$ in the $YZ$‑plane, and $OCP$ in the $XY$‑plane — you get

where $r = |\vec{r}|$. These three cosines are called the direction cosines $l, m, n$ of the vector, and they satisfy $l^{2} + m^{2} + n^{2} = 1$. The coordinates can then be written as $(lr, mr, nr)$, which is a compact way of saying that the vector’s direction is completely specified by its direction cosines, and its length scales them. …

Fig 10.3 is the central diagram for understanding how a vector in 3D space is described by its components and its direction. It shows a point $P(x, y, z)$ in the first octant of a right-handed rectangular coordinate system, with the origin $O$ at the corner of a dashed rectangular box (a cuboid) whose edges lie along the positive $X$, $Y$, and $Z$ axes.

The position vector $\vec{r} = \overrightarrow{OP}$ is drawn as a solid arrow (indigo in the original) from $O$ to $P$, cutting diagonally through the interior of the cuboid. The three edges of the cuboid that meet at $O$ are the coordinate axes; the three edges that meet at $P$ are parallel to the axes. The projections of $P$ onto the axes are labelled $A$ (on the $X$-axis), $B$ (on the $Y$-axis), and $C$ (on the $Z$-axis). The lengths $OA$, $OB$, and $OC$ are exactly the coordinates $x$, $y$, and $z$ of $P$, respectively. So the dashed cuboid makes it visually clear that to go from $O$ to $P$ you can travel $x$ units along $X$, then $y$ units parallel to $Y$, then $z$ units parallel to $Z$ — or take the direct vector $\vec{r}$.

At the origin $O$, three small arcs indicate the direction angles $\alpha$, $\beta$, and $\gamma$ that $\vec{r}$ makes with the positive $X$, $Y$, and $Z$ axes. These are the angles between the vector and each axis, measured in the usual way (between $0^\circ$ and $180^\circ$).

The figure also isolates a right triangle $OAP$, with the right angle at $A$ (where the perpendicular from $P$ meets the $X$-axis). In this triangle, $OA = x$ is the adjacent side to angle $\alpha$, $OP = r$ is the hypotenuse, and $AP$ is the opposite side. From this triangle the textbook derives the first direction cosine:

Similarly, by considering the right triangles $OBP$ (right angle at $B$) and $OCP$ (right angle at $C$), you get:

These three cosines are called the direction cosines of $\vec{r}$, often denoted by $l$, $m$, and $n$ respectively. The magnitude $r = |\vec{r}|$ is given by the distance formula:

Because $x = r\cos\alpha$, $y = r\cos\beta$, $z = r\cos\gamma$, the coordinates of $P$ can be written as $(lr, mr, nr)$ where $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$. The numbers $lr$, $mr$, $nr$ are called the direction ratios of $\vec{r}$, often denoted by $a$, $b$, $c$.