Table of contents  
Vectors and their Representation  
Laws of Addition and Subtraction of Vectors  
Position Vector  
Resolution of Vectors  
Results related to Vector Addition  
Products of Vectors 
It is obvious in scalar algebra, that 3+4 = 7. But if for instance we assume that 3 and 4 are vectors rather than scalars then 3+4 could be anything between 7 to +7. Sounds interesting ! This is the real power of vectors. In this chapter we will study vector algebra and motion in two dimensions.
A quantity is said to be a vector if it satisfies the following conditions.
(a) Obeys the law of parallelogram addition (discussed later in document).
(b) Has a specified direction.
A vector is represented by a line with an arrowhead. The point O from which the arrow starts is called the tail or initial point, or origin of the vector. Point A, where the arrow ends, is called the tip or head or terminal point of the vector. A vector displaced parallel to itself remains unchanged.
(i) Keep vectors s.t. tail of one vector coincides with head of others.
(ii) Join tail of first to head of the other by a line with arrow at head of the second.
(iii) This new vector is the sum of two vectors. (also called resultants)
(i) (ii) (iii)
This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors s.t. tail of next vector lies on head of former.
When we connect the tail of first vector to head off last we get resultant of all the vectors.
[Associative Law]
(i) Keep two vectors such that their tails coincide.
(ii) Draw parallel vectors to both of them considering both of them as sides of a parallelogram.
(iii) Then the diagonal drawn from the point where tails coincide represents the sum of two vectors, with its tail at point of coincidence of the two vectors.
(i) (ii) (iii)
Note: and thus [Commutative Law]
Note: Angle between 2 vectors is the angle between their positive directions.
Suppose angle between these two vectors is θ, and
(AD)^{2} = (AE)^{2} +(DE)^{2}
= (AB + BE)^{2} + (DE)^{2}
= (a +b cosθ)^{2} + (b sinθ)^{2 }
= a^{2} + b^{2} cos^{2}θ + 2ab cosθ + b^{2} sin^{2}θ
= a^{2} + b^{2} + 2ab cosθ
Thus, AD =
or
angle α with vector a is
tan α = =
Important points :
i.e., θ = 0º, i.e. vectors are like or parallel and R_{max} = A + B.
Negative of a vector say is a vector of the same magnitude as vector but pointing in a direction opposite to that of .
Thus, can be written as or is really the vector addition of and .
Suppose angle between two vectors and is θ. Then angle between and will be 180° θ as shown in figure.
Magnitude of will be thus given by
S = =
or S = ...(i)
For direction of we will either calculate angle α or β, where,
tanα = = ...(ii)
or tanβ = = ...(iii)
Ex.3 Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude.
⇒ Sol. =
⇒ α = 30°
[Here shows what is angle between both vectors = 120° and not 60°]
Note: or can also be found by making triangles as shown in the figure. (a) and (b)
Or
Ex.4 Two vectors of equal magnitude 2 are at an angle of 60° to each other find magnitude of their sum & difference.
Sol.
Ex.5 Find and in the diagram shown in figure. Given A = 4 units and B = 3 units.
Sol. Addition :
R =
= = units
tanα = = = 0.472
a = tan^{1}(0.472) = 25.3°
Thus, resultant of and is units at angle 25.3° from in the direction shown in figure.
Subtraction : S =
= =
and tanθ =
= = 1.04
∴ α = tan^{1} (1.04) = 46.1°
Thus, is units at 46.1° from in the direction shown in figure.
Position vector is a vector that gives the position of a point with respect to the origin of the coordinate system. The magnitude of the position vector is the distance of the point P from the origin O. Vector OP is the position vector that gives the position of the particle with reference to O.
If the resultant, then conversely i.e. the vector can be split up so that the vector sum of the split parts equals the original vector If the split parts are mutually perpendicular then they are known as components of and this process is known as resolution.
The orthogonal component of any vector along another direction which is at an angular separation θ is the product of the magnitude of the vector and cosine of the angle between them (θ).
In physics, resolution gives unique and mutually independent components only if the resolved components are mutually perpendicular to each other. Such a resolution is known as rectangular or orthogonal resolution and the components are called rectangular or orthogonal components.
Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure. Let θ be the angle between P and Q and R be the resultant vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q
So, we have
R = P + Q
Now, expand A to C and draw BC perpendicular to OC.
From triangle OCB,
In triangle ABC,
Also,
Magnitude of resultant:
Substituting value of AC and BC in (i), we getwhich is the magnitude of the resultant.
Direction of resultant: Let ø be the angle made by resultant R with P. Then, From triangle OBC,
There are two kinds of multiplication for vectors. One kind of multiplication is the scalar product, also known as the dot product. The other kind of multiplication is the vector product, also known as the cross product. The scalar product of vectors is a number (scalar). The vector product of vectors is a vector.
The scalar product is discussed in detail in the document: Scalar Product of Vector.
The scalar product is discussed in detail in the document: Cross Product of Vector.
102 videos411 docs121 tests

1. What is the significance of a position vector in vector representation? 
2. How are vectors resolved into components using the process of resolution of vectors? 
3. What are the laws governing the addition and subtraction of vectors? 
4. What are the key results related to vector addition that are important to understand? 
5. How are products of vectors calculated, and what are their applications in various fields? 

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