Learn what a vector is,’explore its types and properties
then see how vectors are represented — one step at a time.
Introduction to Vectors
A vector is a physical quantity that has both magnitude (size) and a definite direction. Vectors are used to solve many daily-life problems where both how much and in which direction matter. The idea was first developed by the Irish mathematician William Rowan Hamilton.
Physical quantities that can be measured are of two types:
Scalar quantity – has magnitude only, with no direction (e.g., mass of a pumpkin, time, height of a child, distance).
Vector quantity – has both magnitude and a definite direction (e.g., velocity of a girl going east, acceleration of a falling ball, force, weight).
Section Formula – Coordinates of a Point Dividing a Line Segment
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Scalar
At what temperature does water boil in general? Discuss.
Water boils at 100°C. Here, the magnitude (100) and unit (°C) fully describe the temperature. Therefore, temperature is a scalar quantity.
Physical quantities that have only magnitude are called scalar quantities.
Examples: distance, speed, time, temperature, work, power, energy, etc. are scalar quantities.
If P(x, y) divides the line segment joining points A(x1, y1) and B(x2, y2) externally in the ratio m1:m2, then AP:BP = m1:m2
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Vector
Study the figure and answer:
In the figure, if a girl walks 30 m in a straight line from point O, where will she reach?
While discussing the above question, if she walks 30 m east, she reaches point B. If she walks west, she reaches point C and if she walks northeast, she reaches point A. In each case, the girl moves from point O toward a specific direction and reaches a definite point. This is an example of displacement.
Therefore, the displacement has a magnitude of 30 m but the direction differs according to the point reached.
Therefore, displacement has both magnitude and direction, so it is a vector quantity.
Physical quantities that have both magnitude and direction are called vector quantities.
Examples: displacement, velocity, acceleration, force, weight, etc.
External Division
Suppose, in the figure, the line joining A(x1, y1) and B(x2, y2) is divided externally by P(x, y) in the ratio m1:m2.
Or, APBP = m1m2
Now, draw AL⊥OX, BM⊥OX, PN⊥OX, AR⊥PN and BS⊥PN
In the figure, AR = LN = ON − OL = x − x1,
BS = MN = ON − OM = x − x2
PR = PN − RN = PN − AL = y − y1
PS = PN − SN = PN − BM = y − y2
And APPB = m1m2
Right angled triangles ΔPAR and ΔBSP are similar.
ARBS = APBP[ ∵ The ratio of corresponding sides of similar triangle]
Or, x − x1x − x2 = m1m2
Or, m2(x − x1) = m1(x − x2)
Or, m2x − m2x1 = m1x − m1x2
Or, m1x2 − m2x1 = m1x − m2x
Or, m1x2 − m2x1 = x(m1 − m2)
Or, x = m1x2 − m2x1m1 − m2
Midpoint Formula
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If P(x, y) be the midpoint of the line segment joining the points A(x1, y1) and B(x2, y2), then AP = BP and AP:BP = m1:m2 = 1:1.
Types of Vector
Column Vector
In the given figure, the displacement from $O$ to $R$ is represented by $\overrightarrow{OR}$.
Let $R(3,3)$ be a point in the first quadrant. The position vector of $R$ is $\overrightarrow{OR}=3\hat{i}+3\hat{j}$. Its magnitude is $|\overrightarrow{OR}|=\sqrt{3^2+3^2}=3\sqrt{2}$ and the angle with the positive $X$-axis is $\theta=\tan^{-1}\left(\frac{3}{3}\right)=45^\circ$.
Row Vector
The vector $\vec{a}=\begin{pmatrix}3\\3\end{pmatrix}$ can also be written as: $\vec{a}=(3,3)$. Here, the x-component and y-component are arranged in a row, separated by a comma and enclosed in small brackets. Such vectors are called row vectors.
For example: $\vec{b}=(3,-2)$.
Position Vector
In the adjacent figure, what is the initial point of $\overrightarrow{OA}$?
Does $OA$ represent the position of point $A$? Discuss.
If $\overrightarrow{OA}$ displaces $O(0,0)$ to $A(x,y)$, then $\overrightarrow{OA}$ represents the position of $A(x,y)$. Here, $\overrightarrow{OA}$ is called a position vector.
Here, $\overrightarrow{OA}$ is called a position vector. The initial point of a position vector is always the origin $O(0,0)$.
For example: The position vector of $B(3,2)$ is $\overrightarrow{OB}$, where $\overrightarrow{OB}=(3,2)$.
Thus, a vector starting from the origin is called the position vector of its final point.
Hence, $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are the position vectors of points $A$ and $B$ respectively.
Zero or Null Vector
Suppose two people are pulling the same rope from its two ends with equal force in opposite directions. In this situation, the net force on the rope is zero because the two equal forces act in opposite directions, balancing each other and resulting in no effective force on either side.
If a vector displaces $A(x,y)$ to $A(x,y)$ itself, what will be its magnitude? Discuss.
Here, the magnitude of the applied vector is zero (0). Such a vector with zero magnitude is called a zero vector.
$\vec{a}=(0,0)$ is a zero vector. Since $|\vec{F_1}|=|\vec{F_2}|$, $\vec{a}=(0,0)=$ Zero Vector.
Equal Vectors
In the adjacent figure, find the magnitude and direction of $\overrightarrow{AB}$ and $\overrightarrow{CD}$. Are their magnitudes equal? Are their directions also the same? Discuss in groups and present your conclusion in the classroom.
Here, $\overrightarrow{AB}$ displaces point $A$ to $B$ with a certain magnitude and direction, and $\overrightarrow{CD}$ displaces point $C$ to $D$ with the same magnitude and in the same direction. Vectors that have equal magnitude and the same direction are called equal vectors.
For example: if $\vec{a}=(x_1,y_1)$ and $\vec{b}=(x_2,y_2)$ are equal, then it must be $x_1=x_2$ and $y_1=y_2$.
Negative Vector
In the adjacent figure, find the magnitude and direction of $\overrightarrow{AB}$ and $\overrightarrow{BA}$. Are their magnitudes equal? Are their directions also the same? Discuss in groups and present your conclusion in the classroom.
Here, $\overrightarrow{AB}$ displaces point $A$ to $B$ with a certain magnitude and direction, while $\overrightarrow{BA}$ displaces point $B$ to $A$ with the same magnitude but in the opposite direction. Though their magnitudes (lengths of the line segments) are equal, their directions are opposite.
Vectors having equal magnitudes but opposite directions are called negative vectors. $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are negative vectors of each other. Therefore, $\overrightarrow{AB}=-\overrightarrow{BA}$. Example: If $\vec{a}=(6,2)$ and $\vec{b}=(-6,-2)$, they are negative vectors of each other.
Like and Unlike Vectors
In the figure above, vectors $\vec{a}$ and $\vec{b}$ have the same direction. Thus, vectors having the same direction are called like vectors. For example: $\vec{a}=(6,4)$ and $\vec{b}=(3,2)$ are like vectors because both of them are positive.
But vectors $\vec{c}$ and $\vec{d}$ have opposite directions. Thus, vectors having opposite directions are called unlike vectors. For example: $\vec{c}=(-6,-4)$ and $\vec{d}=(3,2)$ are unlike vectors because one vector is negative and the other is positive.
Like or unlike vectors are parallel or collinear vectors.
Similarly, vectors $\vec{d}$ and $\vec{f}$ have neither the same direction nor opposite directions. Therefore, $\vec{e}$ and $\vec{f}$ are neither like nor unlike vectors.