Vectors in Maths (original) (raw)
Last Updated : 12 Jun, 2026
In mathematics, vectors are fundamental objects that represent quantities with both magnitude and direction. They are widely used in various branches of mathematics, physics, engineering, computer science, and other disciplines.
In the above figure, the length of the line represents the magnitude, while the arrowhead indicates the direction of the vector. A vector can be thought of as a directed line segment denoted as\overrightarrow{\rm AB} where:
- A: Initial point (starting point)****.**
- **B: **Terminal point (endpoint).
**Key Features of Vectors:
**1. Magnitude:
- The size or length of the vector.
- Denoted by |\vec{v}| \text{or} ||\vec{v}||
**2. Direction:
- The direction of the vector in space.
- Represented by the arrowhead that indicates where the vector points.
**Real-life analogy of Vectors
To better understand vectors, consider a situation where a football coach is training a goalkeeper to pass the ball. The coach needs to instruct the goalkeeper:
- **Direction: Where to send the ball (toward another player or a specific region).
- **Magnitude: How hard to kick the ball (the strength of the pass).
In this case, the act of passing the ball combines both magnitude (how strong the kick is) and direction (where the ball should go). Such quantities, which require both magnitude and direction, are called vectors.
Representation of Vector
Vectors are represented by taking an arrow above the quantity to indicate that it has both magnitude and direction. For example:
The Force vectoris represented \vec{F} where the arrow above F represents that it is a vector quantity.
Vectors can also be represented by taking their respective magnitude in x, y, and z-directions.
- The x-direction is shown using \hat{i},
- The y-direction is shown using \hat{j},
- The z-direction is shown using \hat{k}.
Thus, a vector \vec{A} can be written as:
\bold{\vec{A} = x\hat{i} + y\hat{j} + z\hat{k}}
Here:
- **x: Magnitude of the vector along the x-axis,
- **y: Magnitude along the y-axis,
- **z: Magnitude along the z-axis.
**Note: The point where the vector starts is called the tail of the vector and the endpoint of the vector is called the head of the vector. We can also denote the vector as the coordinate point in 3-Dimensions.
Components of Vectors
A vector can be easily broken into its two components which represent the value of the vector in perpendicular dimensions. In a 2-D coordinate system, we can easily break the vector into two components namely the x-component and y-component.
For any vector \vec{A} its,
- x-components is Ax and its value is Ax = A cosθ
- Y-components is Ay and its value is Ay = A sinθ
where **θ is the angle formed by the vector with the positive x-axis. Also, the magnitude of the vector A is calculated using the formula,
|A| = √[(Ax)2 +(Ay)2]
Angle Between Two Vectors
If two vectors in the 2-D plane intersect each other then the angle between them can easily be calculated using the dot product of the vector formula. We know that for two vectors vector a, and vector b their dot product is given by,
\vec{a} \cdot \vec{b} = |a|.|b|.cos θ
We can easily calculate the dot product of the two vectors using the dot product rule and then taking the inverse trigonometric cos function on both sides we can easily calculate the angle between two vectors as,
θ = cos-1[(a · b)/|a||b|]
Important Vector Formulas
Vector uses various formulas to solve complex problems efficiently. These formulas are very helpful in understanding and solving vector-related problems.
**1. Vector Addition and Subtraction
If\vec{A} = (ai + bj + ck)and\vec{B} = (pi + qj + rk)
- **Addition: (ai + bj + ck) + (pi + qj + rk) = (a + p)i + (b + q)j + (c + r)k
- **Subtraction: (ai + bj + ck) - (pi + qj + rk) = (a - p)i + (b - q)j + (c - r)k
**2. Dot Product
- (ai + bj + ck) · (pi + qj + rk) = (a · p)i + (b · q)j + (c · r)k
**3. Cross Product
If \vec{A} = ai + bj + ck and \vec{B} = pi + qj + rk, then
- A × B = (br - cq)i + (ar - cp)j + (aq - bp)k
**4. Angle between two vectors
The angle between two vectors \vec{A} and \vec{B}is:
\theta = cos^{-1} \big(\frac{\vec{A}\cdot\vec{B}}{|\vec{A}||\vec{B}|}\big)
**5. Properties of Vector Multiplication
**Associative Property:
- A · B = B · A
- A × B ≠ B × A
- A × B = -B × A
**6. Other Properties
- i · i = j · j = k · k = 1
- i · j = j · k = k · i = 0
- i × j = k
- j × k = i
- k × i = j
Solved Examples
**Example 1: Find the dot product of vectors P(a, b, c) and Q(p, q, r).
**Solution:
We know that dot product of the vector is calculated by the formula,
P·Q = P1Q1 + P2Q2+……….PnQn
Thus,
P·Q = a·p + b·q + c·r
The dot product of vector P and vector Q is ap + bq + cr (Scalar quantity)
**Example 2: Find the dot product of vectors P(1, 3, -5) and Q(7, -6, -2).
**Solution:
We know that dot product of the vector is calculated by the formula,
P·Q = P1Q1+P2Q2+……….PnQn
Thus,
P·Q = 1·7 + 3·(-6) + (-5)·(-2)
⇒ P·Q = 7 - 18 + 10
⇒ P·Q = 17 - 18
⇒ P·Q = -1The dot product of vector P and vector Q is -1
**Example 3: Let's say two vectors are defined as \bold{\vec{b} = \vec{e} -\vec{c} + 2\vec{d}} and \bold{\vec{a} = 3\vec{e} -\vec{d} + 2\vec{c}}. Find, \bold{\vec{b} + \vec{a}}
**Solution:
Given,
\vec{b} = \vec{e} -\vec{c} + 2\vec{d} ....(1)
\vec{a} = 3\vec{e} -\vec{d} + 2\vec{c} ....(2)
Now, let's calcualte \vec{b} + \vec{a}
\vec{b} + \vec{a} = (\vec{e} -\vec{c} + 2\vec{d}) + (3\vec{e} -\vec{d} + 2\vec{c})
\Rightarrow \vec{b} + \vec{a} = 4\vec{e} +\vec{c} + \vec{d}