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
- 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 vector is 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
\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.
Magnitude of Vectors
The magnitude of a vector represents the strength of the vector. We can calculate the magnitude of the vector by taking the square root of the sum of the squares of each component in the x, y, and z directions.
The magnitude of a vector is calculated by taking the square root of the sum of the squares of the components of the vector in the x, y, and z directions.
For any vector
|\vec{A}| = \sqrt{ a^2 + b^2 + c^2}
The magnitude of a vector is a scalar value.
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
- 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|]
Types of Vectors
Vectors can be classified into different categories on the basis of their magnitude and direction.
The various types of vectors are listed below:
- Zero Vector: A vector with zero magnitude and no direction, written as (0, 0, 0), and satisfies
\vec{a} + \vec{0} = \vec{a} .
- Unit Vector: A vector with unit magnitude, given as
\bold{\hat{c} = \frac{\vec{c}}{|\vec{c}|}} .
- Equal Vector: Vectors with the same magnitude and direction, regardless of their initial and terminal points.
- Negative Vector: A vector with the same magnitude but opposite direction as the original vector, denoted as
-\vec{A} .
- Co-Initial Vectors: Vectors that originate from the same point are called co-initial vectors.
- Collinear Vectors: Two vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
- Parallel Vectors: Vectors with the same direction and an angle of zero between them, though magnitudes may differ.
- Orthogonal Vectors: Vectors at a right angle (90°) to each other, with a dot product of zero.
Operations on Vectors
We can perform various operations on Vectors by taking a geometrical approach or by taking a coordinate system approach. Various operations in vector are,
Addition of Vectors: The addition of two vectors
\vec{A} + \vec{B} = (Ax + Bx, Ay + By, Az + Bz)
Subtraction of Vectors: The difference between two vectors is given by subtracting their corresponding components:
\vec{A} - \vec{B} = (Ax - Bx, Ay - By, Az - Bz)
Multiplication of a Vector by Scalar: Multiplying a vector
K\vec{A} = (kAx, kAy, kAz).
Dot Product: The dot product of two vectors is a scalar:
\vec{A} \cdot \vec{B} = AxBx + AyBy + AzBz
Cross Product: The cross product of two vectors is a vector perpendicular to both:
∣\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}
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
- 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
- A × B = (br - cq)i + (ar - cp)j + (aq - bp)k
4. Angle between two vectors
The angle between two vectors
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 on Vectors in Maths
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
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}
Example 4: Find the magnitude of the vector A = 2i - 5j + 4k, using vector algebra.
Solution:
Given Vector,
Vector A = 2i - 5j + 4k
We know that magnitude of the vector A is |A| i.e.
|A| = √ (a2+b2+c2)
⇒ |A| = √ (22+(-5)2+42)
⇒ |A| = √(4 + 25 + 16)
⇒ |A| = √(45) = 3√(5)