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Vector is a mathematical and scientific quantity that has both magnitude and direction. It has more than one value, such as momentum, displacement, weight, acceleration, force, velocity, etc.
- Vectors can be represented by a direct line segment.
- It is also used for tuples, a finite sequence of fixed length numbers.
- The direction starts from its tail to its head.
- Length represents the magnitude of the vector with an arrow that indicates the direction.
- In Maths and Science, there are two types of quantities: Scalars and Vectors.
- Scalar quantities are those which only have magnitude or a value.
- Some examples of scalar quantities include speed, area, mass, etc.
- Vectors were first used to explain the concept of electromagnetic induction.
Read More: Scalar triple product of vectors
Key Terms: Vectors, Line Segment, Scalar Quantity, Magnitude, Direction, Position Vector, Collinear Vector, Zero Vector, Unit Vector, Coplanar Vector
What are Vectors?
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Vectors are quantities having magnitude as well as direction. They are the objects which are found in the accumulated form in vector spaces accompanying two types of operations.
- These operations include the addition of two vectors and the multiplication of the vector.
- The operations can alter the proportions and order of the vector, but the result still remains in the vector space.
- Vectors are also known as Geometric vectors, Euclidean vectors or Spatial vectors.
- It is often recognized by symbols such as U, V, and W.
- Two vector quantities are said to be equal if they have the same direction and magnitude.
- It is a topic covered in Class 12 Maths.
Solved Example of VectorsExample 1: Find the magnitude of the vector a = 4i - 3j + k, using the formula from vector algebra. Solution: The given vector is a = 4i - 3j + k. The magnitude of the vector is |a| = √42+(−3)2+12 √16+9+1=26 Therefore, the magnitude of the vector is 26. |
Read More: Signum Function
Representation of Vectors
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A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector.
- It is represented by a or commonly as AB.
- In this line segment AB, A is the starting point, and B is the terminal point of the line.
- Addition of two vectors can be represented as
![sum]()
- It also helps in solving the angles between vectors and calculate their areas and volumes.
Representation of vector
Read More:
| Chapter Related Topics | ||
|---|---|---|
| Vector product of two vectors | Cross Product | Real Valued Functions |
| Angle of Depression | Angle Between Two Vectors | Factorial Formula |
Magnitude of Vectors
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The magnitude of AB or a is simply the length of the vector. The magnitude is represented by |AB| or |a| or simply a. Here, the magnitude of any vector, i.e. |a|, is never less than zero.
- Magnitude of any vector is its length, and length is never negative.
- It is calculated using the Pythagoras theorem.
- It is the square root of the sum of the squares of a vector components.
- The magnitude is a scalar value.
Magnitude of a vector
Position Vectors
Position vectors represents a straight line with one end fixed to a body and the other end attached to a moving point. It is basically used to describe the position of the point relative to the body.
- The position vector will change in length or in direction or in both of them with the movement of the point.
- If P is a point in space having its coordinates as (x,y,z) with respect to the origin (0,0,0).
- |OP| = \(\sqrt{x^2 + y^3 + z^2}\)
Position Vector
Read More: Calculus Formula
Direction Cosines
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A directed line segment AB makes 3 angles with the AM, AN and AO coordinate axes respectively, then cosα, cosβ and cosγ are called as the direction cosines of the line segment AB. These angles are denoted by x, y and z.
x= cosα , y= cosβ and z= cosγ.
- Let us suppose that x, y and z are the direction cosines of a line segment.
- l, m and n be three numbers.
- xl=ym=zn= r, where x2+y2+z2=1
Relationship between Direction Ratios, Magnitude and Direction Cosines of a Vector
The relation between the direction ratios (x,y,z), magnitude(r) and direction cosines(l,m,n) of a vector can be written as:
l=(xa), m=(ya) and n=(za)
For more information, check this.
Read More: Midpoint Formula
Types of Vectors
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The different types of vectors are as follows:
Negative Vectors
A vector is said to be a negative vector if they have same magnitude but have different direction.
Collinear Vectors
A vector is said to be collinear vector if they have same direction but have different magnitude. It is also called as parallel vectors.
Zero Vectors
A vector is said to be zero vectors that has zero magnitude and no direction. It is denoted by (0,0,0). It is also known as additive identity of vectors.
Coplanar Vector
A vector is said be a coplanar vectors which consists of three vectors lying in the same plane or parallel lines.
Unit Vector
A vector is said to be a unit vector that has a magnitude value equivalent to one. It is also known as multiplicative identity of vectors.
Co-Initial Vector
A vector is said to be a co-initial vector if two vector have same starting point.
The video below explains this:
Types of Vectors Detailed Video Explanation:
Operations on Vectors
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The various operations of vectors are as follows:
Addition of Vectors
The two vectors a and b are added by giving the value of sum equivalent to a + b. The addition of vectors can be done using two laws: triangle law of vector addition and parallelogram law of vector addition.
Triangle Law of Vector Addition
Triangle law of vector addition refers to the two vectors that are represented as two sides of the triangle with the order of direction and magnitude.
- The third side of the triangle implies the magnitude and direction of the resultant vector.
- The initial point of one vector must coincide with the ending point of another vector.
Triangle law of vector addition
Parallelogram Law of Vector Addition
The parallelogram law of vector addition states that two vectors act along the adjacent side of a parallelogram. In this law, magnitude is equal to the length of the sides.
- Both vectors point away from the common vertex.
- It result in the formation of diagonal of parallelogram that passes through the common vertex.
Parallelogram law of vector addition
Properties of Addition of Vectors
- Commutative: For vectors x and y, we can say, x + y = y + x
- Additive Identity: Zero Vector is an additive identity for any vector x where x + 0 = x
- Additive Inverse: Negative vector is an additive inverse for any vector x where x + (-x) = 0
- Associative: For vectors x, y and z, we can say, x + (y + z) = (x + y) + z
Scalar Multiplication of Vectors
When a vector is multiplied by a scalar quantity, then the result will be a vector quantity. If x is a given vector and is a scalar quantity, then the multiplication will be x, which will also be a vector quantity.
- It will be collinear to the x vector, and its magnitude is k times of vector x.
- Its direction will be the same as the vector.
Properties of Scalar Multiplication of Vectors
Consider two vectors x and y and two scalar quantities a and b, the multiplication of vectors will be done as follows:
- a(b x) = (a b) x
- a(x + x) = a x + a x
- (a + b) x = a x + b x
Read More: Integrals
Applications of Vectors
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The important applications of vectors are as follows:
- The concept of vectors is used in the field of quantum mechanics.
- It is used in the field of engineering where force is stronger than the structure.
- The concept is the propagation of waves, which includes vibration propagation, sound propagation and AC wave propagation.
- In fluid mechanics, the velocity of a pipe is found in terms of vectors.
- It helps in calculating the value of force applied in three directions.
Read More:
| Class 12 Maths Related Concepts | ||
|---|---|---|
| Methods of Integration | Vector Formula | Differential Equations |
Important Topics for JEE MainAs per JEE Main 2024 Session 1, important topics included in the chapter vectors are as follows:
Some memory based important questions asked in JEE Main 2024 Session 1 include:
|
Things to Remember
- Vector quantities involve more than one value, such as momentum, displacement, force and velocity.
- Geometrically, it can be illustrated as a directed line segment.
- A line having an arrowhead is known as a directed line.
- A segment of the directed line has both direction and magnitude.
- This segment of the directed line is known as a vectors.
Sample Questions
Ques. Given vector V, having a magnitude of 20 units & inclined at 60°. Break down the given vector into its two components. (3 marks)
Ans. Given, Vector V having magnitude|V| = 20 units and θ = 60°
Horizontal component (Vx) = V cos θ
Vx = 20 cos 60°
Vx = 20 × 0.5
Vx = 10 units
Now, Vertical component(Vy) = V sin θ
Vy = 20 sin 60°
Vy = 20 × √3/2
Vy = 10√3 units
Ques. Find the magnitude of vector a (5, 12). (2 marks)
Ans. Given Vector a = (5,12)
|a|= √(x2+y2)
|a|= √(52+122)
|a|= √(25+144) = √169
Therefore, | a |= 13
Ques. Given vector V, having a magnitude of 30 units & inclined at 30°. Break down the given vector into its two components. (3 marks)
Ans. Given, Vector V having magnitude|V| = 30 units and θ = 30°
Horizontal component (Vx) = V cos θ
Vx = 30 cos 30°
Vx = 30 × √3/2
Vx = 15√3 units
Now, Vertical component(Vy) = V sin θ
Vy = 30 sin 30°
Vy = 30 × 0.5
Vy = 15 units
Ques. How is scalar quantity different from vector one? (2 marks)
Ans. The main difference between scalar and vector quantity is that the scalar quantity has only magnitude, whereas the vector quantity has both magnitude and direction. It is a vector quantity if a quantity has a direction associated with it. On the other hand, a scalar quantity is not associated with the direction. Moreover, vector, as it specifies both direction as well as magnitude, is a vector quantity but speed is a scalar quantity.
Ques. Find a vector in the direction of vector 
which has a magnitude 21 units. (5 marks)
Ans. In order to find a vector in the direction of a given vector, first of all we find the unit vector in the direction of direct vector and then multiply it with the given magnitude.
The unit vector in the direction of the given vector 
is
Therefore the vecror f magnitude equal to 21 units and in the direction of is,
Ques. Find the value of p for which the vectors 
and 
are parallel. (3 marks)
Ans. We have and which are parallel vectors, so their directionl ratios will be proportional.
Ques. What is the value of cosine of the angle which the vector 
makes with Y- axis. (4 marks)
Ans. It is given,
Now, unit vector in the direction of is
Therefore, the cosine of angle that the given vector makes with Y-axis is 1/√3.
Ques. Determine a vector of magnitude 5 units and parallel to the resultant of 
and 
. (5 marks)
Ans. Firstly, find the resulatnt of the vectors and 
, 
. Then we have to find the unit vector in the direction of i.e., the unit vector is multiplied by 5.
Given, and
Now, resultant of the above vectors =
Ques. Let 
and 
. Find out a vector of magnitude 6 units, which is parallel to the vector 
. (5 marks)
Ans. Firstly, we have to find the vector , then find the vector in the direction of , i.e., the unit vector multiplied by 6.
Given, ;
and
Therefore,
Now, a unit vector in the direction of vector
Ques. Find the position vector of a point R, which divides the line joining two points P and Q and whose position vectors are 
and 
respectively, externally in the rartio 1:2. Also, show that P is the mid-point of the line segment RQ. (5 marks)
Ans. Given, 
= position vector of P =
and 
= position vector of Q =
Let 
be the position vector of the point R which, divides PQ in the ratio 1: 2 externally
Now, it must be shown that P is the mid oint of RQ.
Therefore, P is the mid-point of the line segment R.
Ques. Given vector V, having a magnitude of 40 units & inclined at 90°. Break down the given vector into its two components. (3 marks)
Ans. Given, Vector V having magnitude|V| = 40 units and θ = 90°
Horizontal component (Vx) = V cos θ
Vx = 40 cos 90°
Vx = 40 × 0
Vx = 0 units
Now, Vertical component(Vy) = V sin θ
Vy = 40 sin 90°
Vy = 40 × 1
Vy = 40 units
Ques. Find the magnitude of vector a (6, 8). (2 marks)
Ans. Given Vector a = (6,8)
|a|= √(x2+y2)
|a|= √(62+82)
|a|= √(36 + 64) = √100
Therefore, | a |= 10
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