Introduction to Vectors | Mathematics (Maths) for JEE Main & Advanced PDF Download

A vector has direction and magnitude both but scalar has only magnitude. 

• Magnitude of a vector a is denoted by |a| or a. It is non-negative scalar.
• Direction of a vector is the angle made by the vector with the horizontal axis, that is, the X-axis. The direction of a  vector formula is related to the slope of a line. We know that the slope of a line that passes through the origin and a point (x, y) is y/x. We also know that if θ is the angle made by this line, then its slope is tan θ, i.e., tan θ = y/x. Hence, θ = tan^-1 (y/x). 

Equality of Vectors

Two vectors a and b are said to be equal written as a = b, if they have (i) same length (ii) the same or parallel support and (iii) the same sense.

Types of Vectors

(i) Zero or Null Vector

A vector whose initial and terminal points are coincident is called zero or null vector. It is denoted by 0.

(ii) Unit Vector 

A vector whose magnitude is unity is called a unit vector which is denoted by n^ˆ

(iii) Negative of a Vector 

A vector having the same magnitude as that of a given vector a and the direction opposite to that of a is called the negative of a and it is denoted by —a.

(iv) Like and Unlike Vectors 

Vectors are said to be like when they have the same direction and unlike when they have opposite direction.

(v) Collinear or Parallel Vectors 

Vectors having the same or parallel supports are called collinear vectors.

(vi) Coinitial Vectors 

Vectors having same initial point are called coinitial vectors.

(vii) Localized Vectors 

A vector which is drawn parallel to a given vector through a specified point in space is called localized vector.

(viii) Coplanar Vectors 

A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Otherwise they are called non-coplanar vectors.

Example: In the figure given below, identify Collinear, Equal and Coinitial vectors:

Solution: By definition, we know that

• Collinear vectors are two or more vectors parallel to the same line irrespective of their magnitudes and direction. Hence, in the given figure, the following vectors are collinear:
• Equal vectors have the same magnitudes and direction regardless of their initial points. Hence, in the given figure, the following vectors are equal:
• Coinitial vectors are two or more vectors having the same initial point. Hence, in the given figure, the following vectors are coinitial:

Example: In the given figure, identify the following vectors

1. Coinitial
2. Equal
3. Collinear but not equal

Solution:

• Coinitial vectors have the same initial point. In the figure given above, vectors   are the two vectors which have the same initial point P.
• Equal vectors have same magnitudes and direction. In the figure given above, vectors  are equal vectors.
• Collinear vectors are two or more vectors parallel to the same line. In the figure given above, vectors  are parallel and hence, collinear. Also,  vectors  are parallel and hence, collinear. We know that vectors  are also equal. Hence, vectors are collinear but not equal.

Addition of Vectors

Let a and b be any two vectors. From the terminal point of a, vector b is drawn. Then, the vector from the initial point O of a to the terminal point B of b is called the sum of vectors a and b and is denoted by a + b. This is called the triangle law of addition of vectors.

Parallelogram Law

Let a and b be any two vectors. From the initial point of a, vector b is drawn and parallelogram OACB is completed with OA and OB as adjacent sides. The vector OC is defined as the sum of a and b. This is called the parallelogram law of addition of vectors.

The sum of two vectors is also called their resultant and the process of addition as composition.

Properties of Vector Addition

(i) a + b = b + a (commutativity)

(ii) a + (b + c)= (a + b)+ c (associativity)

(iii) a+ O = a (additive identity)

(iv) a + (— a) = 0 (additive inverse)

(v) (k₁ + k₂) a = k₁ a + k₂a (multiplication by scalars)

(vi) k(a + b) = k a + k b (multiplication by scalars)

(vii) |a+ b| ≤ |a| + |b| and |a – b| ≥ |a| – |b|

Difference (Subtraction) of Vectors

If a and b be any two vectors, then their difference a – b is defined as a + (- b).

Multiplication of a Vector by a Scalar

Let a be a given vector and λ be a scalar. Then, the product of the vector a by the scalar λ is λ a and is called the multiplication of vector by the scalar.

Important Properties

(i) |λ a| = |λ| |a|

(ii) λ O = O

(iii) m (-a) = – ma = – (m a)

(iv) (-m) (-a) = m a

(v) m (n a) = mn a = n(m a)

(vi) (m + n)a = m a+ n a

(vii) m (a+b) = m a + m b

Vector Triple Product

Vector Triple Product is a concept in vector algebra that involves taking the cross product of three vectors. To find its value, you calculate the cross product of one vector with the cross product of the other two vectors. The result is a new vector.  

Consider next the cross product of  

This is a vector perpendicular to both a   is normal to the plane of   so must lie in this plane. It is therefore expressible in terms of  in the form   To find the actual expression for   consider unit vectors j^{^} and k^{^} the first parallel to  and the second perpendicular to it in the plane  

In terms of  j^{^} and k^{^} and the other unit vector î of the right-handed system, the remaining vector  be written   Then    and the triple product

This is the required expression for in terms of

Similarly the triple product    ...(2)

It will be noticed that the expansions (1) and (2) are both written down by the same rule. Each scalar product involves the factor outside the bracket; and the first is the scalar product of the extremes.

In a vector triple product the position of the brackets cannot be changed without altering the value of the product. For  is a vector expressible in terms of    is one expressible in terms of The products in general therefore represent different vectors. If a vector r is resolved into two others in the plane of    one parallel to and the other perpendicular to it, the former is    and therefore the latter  

Geometrical Interpretation of  

Consider the expression which itself is a vector, since it is a cross product of two vectors   Now is a vector perpendicular to the plane containing   vector perpendicular to the plane   therefore  is a vector lies in the plane of and perpendicular to a . Hence we can express   in terms of  i.e.   where x & y are scalars.

Vector Triple Product Formula

The vector triple product formula can be written as:

Example: Find a vector  and is orthogonal to the vector   It is given that the projection of 

Solution:  A vector coplanar with  is parallel to the triple product,

Example: ABCD is a tetrahedron with A(–5, 22, 5); B(1, 2, 3); C(4, 3, 2); D(–1, 2, –3). Find  What can you say about the values of   Calculate the volume of the tetrahedron ABCD and the vector area of the triangle AEF where the quadrilateral ABDE and quadrilateral ABCF are parallelograms.

Solution:

Example:  Let a x b=c, b x c=a, and a, b, c be the moduli of the vectors a, b, c, then find a and b.

Solution: a = b × c and a × b = c

∴ a is perpendicular to b and c, and c is perpendicular to a and b.

a, b, and c are perpendicular to each other

Now, a = b × c = b × (a × b) = (b . b) a − (b . a) b or 

a =b² a − (b.a) b= b² a, {because a⊥b}

⇒1= b .Therefore,  𝑐 = 𝑎×𝑏 = 𝑎𝑏𝑠𝑖𝑛900ń

Taking the moduli of both sides, c = ab, but b = 1 ⇒ c = a.

Example: Given these simultaneous equations for two vectors x and y.

x + y = a …..(i)

x × y = b …..(ii)

x . a = 1 …..(iii)

Find the values of x and y.

Solution:  By multiplying (i) scalarly by a, we get

a . x + a . y = a²

∴ a . y = a² − 1 ..(iv),

{By (iii)} Again a × (x × y) = a × b or (a . y) x − (a . x) y = a × b

(a² − 1) x − y = a × b ..(v),

Adding and subtracting (i) and (v),

we get x =  𝑎+(𝑎×𝑏) / [a2] and y = a − x

Applications of Vector Triple Product

The vector triple product isn't just a mathematical curiosity; it finds practical applications in various fields:

• Classical Mechanics: It helps calculate the torque acting on a rigid body and analyse the motion of charged particles in magnetic fields.
• Electromagnetism: It comes in handy when dealing with electromagnetic fields and their interactions with matter.
• Crystallography: It plays a crucial role in understanding the arrangement of atoms in crystals and predicting their properties.

Product of Four Vectors

(a) Scalar Product of Four Vectors: The products already considered are usually sufficient for practical applications. But we occasionally meet with products of four vectors of the following types. Consider the scalar product of  This is a number easily expressible in terms of the scalar products of the individual vectors. For, in virtue of the fact that in a scalar triple product the dot and cross may be interchanged, we may write

 Writing this result in the form of a determinant,

we have

(b) Vector Product of Four Vectors:

Consider next the vector product of  This is a vector at right angles to     and therefore coplanar with   Similarly it is coplanar with    It must therefore be parallel to the line of intersection of a plane parallel to  with another parallel to  

To express the product in      in terms of   regard it as the vector triple product of and  

Similarly, regarding it as the vector product of    we may write it

Equating these two expressions we have a relation between the four vectors

...(3)

Example: Show that ,

Sol.

Example: Show that 

Sol:

Vector Equations

Example: Solve the equation

Sol. From the vector product of each member with a, and obtain

Example: Solve the simultaneous equations

 Sol. Multiply the first vectorially by
which is of the same form as the equation in the preceding example.

Thus  
Substitution of this value in the first equation gives 

Example:  

Sol. Multiply scalarly by

Example:   then prove that

Sol. 

 ...(1)

Solving (2) and  simultaneously we get the desired result.

Example: Solve the vector equation in 

Sol. Taking dot with a = ...(1)

Taking cross with a = ...(2)

Example: Express a vector as a linear combination of a vector  and another perpendicular to A and coplanar with and .

Sol.  is a vector perpendicular to  and coplanar with  and .

Hence let, 

 ...(1)

taking dot with  

again taking cross with