(For HSC & Pre-Admission)

Higher Math 1st Paper

Chapter-02 : Vector
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Dear Students,
You are stepping into a significant part of your educational life. The higher secondary
curriculum is much different and vast compared to the secondary curriculum. The
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Udvash Math Team

 Index
Higher Math 1st Paper
Chapter-02 : Vector
Sl Topic Page
01 Quantity 2
02 Types of vector quantities or different types of vectors 5
03 Addition of Vectors 7
04 Vector Subtraction 13
05 Internal and external division of a line segment between two points 14
06 Geometric proofs using vector addition-subtraction concepts 16
07 Components of a vectors 21
08 Projection & Component (Longitudinal) 23
09 Representation of a vectors in two dimensional cartesian co-ordinate 25
10 Representation of a vector in three dimensional cartesian co-ordinarinates 26
11 Problems related to addition, subtraction and quantification of vectors 28
12 Determination of unit vector towards, to the opposite or to the parallel of a vector 30
13 Multiplication of Vectors 33
14 Multiplication of vectors by a scalar quantity 33
15 Dot product of vectors (scalar multiplication) 35
16 Scalar product of vectors and problems related to two perpendicular vectors 37
17 Problems related to another vector in the same plane as two other vectors 39
18 Problems related to determination of included angle between two vectors 41
19 Determination of perpendicular projection and component of vectors 42
Exercise-2.1
20 Vector/Cross Multiplication/Product 54
21 Problems related to Vector Cross Product and Two Parallel Vectors 60
22 Unit vector perpendicular to the plain formed by two vectors 61
23 Some information related to area 62
24 Some information related t Problems related to determination of the area of polygons 63
25 Some information related to area 64
26 Vector of straight line and cartesian equations in a three dimensional co-ordinate system 70
27 Vector Equation of A Straight Line Passing Through Two Fixed Points 72
28 Determination of the distance between two points on a straight line 74
29 Determination of the included angle between two straight lines 74
30 Determination of the intersecting point from the vector equation of two straight lines (if exists) 75
Exercise-2.2
31 Brainstorming Question 82
32 All Important Formulae in a Body 83
33 Important Practice Problem 85
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 We are familiar with origami or paper crafts since childhood. Even if we can't
build anything else, we can all build boats and planes. In our childhood we used
to make boat and float it in a big bowl. But I am sure that all of us loved to fly
the plane after making it . Ayon and Antor are such children .One day they made
a plane and went to field to fly it .Ayon is standing infront of Antor facing him
keeping a little bit distance . Aim of Antor is to fly up the plane and send it to
Ayon.
But the problem is, the wind is blowing between them like in the given
picture. Now the question is, if Antor throws the plane directly
towards Ayon, then will the plane reach to Ayon? What do you think
? Keep thinking. Within this time, let us go and know about vectors.
It may happen that the answer is hidden within there.

Brief History
The concept of vector is very important in mathematics, physics and engineeri ng. It is such a geometric concept
that has both magnitude and direction. The concept of vector has came to its present state through about 200 years
of gradual changes.

The earliest traces of work with vectors date back to the early 19 th century. In 1827, August Ferdinand Mobius
published a book named Barycentric calculus in which he introduced the concept of Directed line segments
which are denoted by English alphabet. Later, in the late 19 th century, Josiah Williard Gibbs and Oliver Heaviside
elaborated the concept of vector analysis and vector algebra separately. They did this to explain the first principle
of electromagnetism given by James Cleark Maxwell. Josiah Williard Gibbs was a physicist. He found that his
works on physics will become much easier by vector analysis. Later, Oliver Heaviside (1850-1925), another
physicist, also played a role in explaining vector analysis. To understand what vector is, we need to understand
some necessary quantities. Let s discuss about them now.

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 Quantity
Quantity: In this materialistic world the things which can be measured is known as quantity.
Example: Length, mass, force, speed, velocity etc.
Classification of Quantities: Quantities can be classified on different basis. To measure a quantity, it is
needed to measure some fundamental quantities. Based on this, quantities are of 2 types:
(i) Fundamental Quantities
(ii) Derived Quantities
(i) Fundamental Quantities: Imagine I asked you, what is the distance between your house and your
college? You replied that 1 km; How much time it would take to go there? You said 12 minutes. You only
need one piece of data to answer these two questions. In 1 st case how much distance in km and in 2nd case
how much time in minutes. These are fundamental quantities.

The quantities that are independent or neutral or the quantities that does not require any other quantity to
express itself, are called fundamental quantities.

There are total 7 fundamental quantities in the concern of science:

Fundamental Quantities Unit Unit signal
(i) Length Meter m
(ii) Mass Kilogram kg
(iii) Time Second s
(iv) Temperature Kelvin K
(v) Electric current Ampere A
(vi) Luminous intensity Candela cd
(vii) Amount of Substance Mole mol

(ii) Derived Quantities: Now imagine you are going to college by cycle. In cycle s speedometer your speed is
showing 4 ms−1 . That means, your cycle is crossing 4 m distance in 1s. Now observe, for measuring the
speed, Speedometer has to measure 2 quantities. One of them is distance and the other one is time. Here
speed is dependent on other two quantities or it is a derived quantity.

2
The quantities that are dependent on the fundamental quantities or the quantities that require more than one
fundamental quantity to express itself are called derived quantities.
distance
Example: Force = [mass × ,
time2
distance
Speed/velocity = [ ,
time
distance
Work = [ mass × , Charge = [current × time] etc.
time

When I asked you some time ago how far is the college
from your house, you said 1 km. which made me
understand how far your college is. I don't need any
more information. But now if I want to go to your
college, is 1km distance information enough for me?
No, it is not. Because you didn't tell me which direction
to go for 1 km. But if you tell me that I have to go 1 km
east then I will reach your college just fine.

That means for describing some quantities it is enough to know about only the magnitude (like distance).
These are called scalar quantities. But there are some other quantities which can t be expressed completely
only with the value, direction is also needed. Like in the previous example, in a 1 km east direction my
displacement is needed, only then i will be able to reach your college. This displacement is vector quantities.
That is, on the basis of direction, quantities can be divided into 2 types-
(i) Scalar Quantity
(ii) Vector Quantity

(i) Scalar Quantity: A scalar quantity is a quantity that can be expressed entirely by the value alone and
changes only when the value changes.
Example: Length, Distance, Time, Volume etc

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 (ii) Vector Quantity: Quantities that require both magnitude and direction to be fully expressed and
quantities for which, change in only magnitude or direction or both changes the quantity, are called
vector quantities.

Example: Displacement, Force, Acceleration etc.

Expression of vector quantities:

Taking one end of a straight line as the origin and the other end as the endpoint, the directed line segment
expresses a vector quantity. The length of the line indicates the magnitude of the vector quantity, and the
direction indicates the direction of the vector quantity. A vector quantity can be expressed in any of the

following ways in writing: AB , AB, AB

When typing on a computer, AB can also be expressed in bold letters (AB) to represent vector quantities.

Also, the value of a vector quantity is generally expressed in the form of AB or |AB|.

Support Line:
The segment of an infinite straight line directed by a vector is called its support line. [That is, the line
containing a vector is called the support line]

In figure AB vector is a segment of an infinite straight line a portion of which CD is indicated in figure.

Direction:

In figure the direction of vector AB is from point A to point B. Similarly, the direction of BA is from point
B towards point A.

Warning!

Some students want to express BA as ശAB, which is not acceptable.

4
 Types of vector quantities or different types of vectors
(i) Equal Vectors:

AB CD EF
(i) Values are equal. [Length of line segment indicating
direction is equal]
(ii) The support lines are same or parallel.
(iii) Direction is same.

(ii) Opposite Vectors:

AB CD or, EF CD
(i) Values are equal. [Length of line segment indicating
direction is equal]
(ii) The support lines are same or parallel.
(iii) Directions are opposite.
Remember: AB DC

Note
Here, AB is opposite vector of CD, but equal to vector DC.

(iii) Collinear or Parallel Vectors:

(i) Values are equal/not equal.

(ii) Having a parallel support line of a straight line.
(iii) Direction maybe same or opposite.

Collinear or Parallel Vectors are of 2 types:

(a) Like Vector: (b) Unlike Vectors:

(i) Values are same or different. (i) Values are same or different.
(ii) Support line same or parallel. (ii) Support line same or parallel.
(iii) Same direction. (iii) Opposite direction.

5
(iv) Proper Vector: All vectors are proper vectors except the zero vector.
(v) Zero or Null or Improper Vector: A vector whose value is zero is called a null vector.

Note
Direction of null vector is unspecified.

(vi) Unit Vector: A vector whose value is one is called a unit vector. Dividing a vector by its magnitude gives
a unit vector in the direction or parallel to that vector. Unit vector of vector A is usually expressed as a.

AB 5a ⇒ a

∴a
| |

(vii) Rectangular Unit Vectors:

The three unit vectors along the three axes x, y and z in the three
dimensional Cartesian coordinate system are collectively called
rectangular unit vectors.
(i) Unit vector along x-axis:
(ii) Unit vector along y-axis: j
(iii) Unit vector along z-axis: k

(viii) Position Vector:

In three dimensional coordinate system, a vector expressing the position of
a point with respect to the origin of the frame of reference is called a
position vector. The position vector is also sometimes called the radius
vector r . For example, we can express the position of point P in the figure
by the vector OP. ∴ OP r is the position vector or radius vector of point
P. That is, the magnitude of the position vector of a point is the length and
direction is along the line connecting that point from the origin.
OP r

Note
In two-dimensional coordinate geometry, a point is expressed by abscissa(x) & ordinate(y). A point in vector is
denoted by position vector.

(ix) Co-planar Vector:

If the lines containing the vectors lie in the same plane [or lie in a plane parallel to the same plane] then
they are called co-planar vectors.

The vectors indicated in the figure are Co-planar vectors.

6
(x) Reciprocal Vector:
If the value of one of two parallel vectors is reciprocal of other, then they are called reciprocal vectors.
1
Let, AB 5a and CD a [a is a unit vector ]
5
∴ AB and CD are mutually reciprocal vectors.

(xi) Free Vector:

The vector that has a fixed modulus and direction but no fixed position, i.e. if the position of vector can be
moved without changing the modulus or direction then it is called free vector.

Angle Between Two Vector:

To determine the angle between two vectors, α, place the two vectors at a point and extend if necessary, so
that the vectors appear to emanate from the point [O]. In this case the angle between them is the angle
between the two vectors.
Limit of angle: 0 ≤ α ≤ π

Note
α 0 Both vectors are like parallel vectors
α π Both vectors are unlike parallel vectors
α Both vectors are perpendicular to each other

Addition of Vectors
Suppose you have a mass measuring device at home. Again you have 3 bottles full of water. One of 1 L, another
of 2 L and another of 5 L volume. The mass of the bottles is negligible. Now if you put the water bottle of 1 L
volume in the mass measuring machine, you will see that the mass will be shown 1 kg. Then if the 2 L water bottle
is also placed on the machine, the mass will be shown 1 2 3 kg Again, if the 5 L bottle is also placed on
the machine, then the mass will be shown-
1 2 5 8 kg Note: Mass of 1L pure water is 1kg

Note that mass is a scalar quantity and in case of addition, masses or scalar quantities are generally added.
That is, scalar quantities can be added by general algebraic rules.
But suppose you went 3 m east from your house and then 4 m north from there, now if you are asked how
many meters are you away from your house, will your answer be 3 4 7m?
7
 The answer is no, 7 m won't be the right answer because you have to think about direction here as well.
From the figure it can be seen that you are now √3 4 5 m away from your house. In other words,
your displacement here is 5 m, that is, vectors cannot be added according to normal algebraic rules. It
requires vector algebra.

Warning!
In the case of addition of two vector quantities, the two quantities must be of same type. For example, only
velocity can be added to velocity but addition of displacement to velocity is meaningless. The same is true for
scalar quantities. Adding time to mass is meaningless.

The following methods are used for the addition of two vector quantities:
(i) Initial & Terminal Point Rule
(ii) Law of Triangle
(iii) Law of Parallelogram
(iv) Law of Polygon

Initial & Terminal Point Rule:

In the figure, BC CA BA a b

If a and b are two same type of vector quantities, then for addition of a & b vectors:
(i) Let vector BC is taken equal in magnitude and parallel to the vector a.
(ii) Let us draw a vector CA parallel and equal to b at the vertex C of BC (or a).
(iii) Then the vector BA obtained by adding the origin B of BC and the vertex A of CA is the sum or
resultant of a and b.

8
 In simple words if you go from B to C and then from C to A then your resultant displacement is BA. A
triangle is obtained from the intial-terminal point rule through which the triangle formula can be derived.
For ease of understanding we will read triangle formula/rule and formula of parallelogram together.

Law of triangle: Law of Parallelogram:

Suppose a person hits a football with a force FA Suppose, two persons strike a ball with forces FA &
at point A and it stops moving from point A to FB at an angle of α at point O. The ball tends for
point B. The ball stops at point C when another displacement OA for force F and OB for force FB .
person hits it with force FB . That is, the resultant But since two forces act together, the ball will move
displacement of football is AC. along the middle path OC.
Conditions: Conditions:
(i) Two forces (vectors) act at different points (i) Two vectors/forces act at the same point O.
(ii) Two forces (vectors) act at different times (ii) Two vectors/forces act simultaneously.
(iii) The forces/vectors can be represented in the (iii) Two adjacent sides of a parallelogram (OA and
same order by two adjacent sides AB and OB) will represent magnitude and direction of the
BC of a triangle. [anti-clock wise] forces/vectors.
Result: Result:
Then the third side of that triangle AC indicated Then the diagonal (OC) along that point of the
in reverse [clock wise] will represent value and parallelogram will indicate the magnitude and
direction of the resultant. direction of the resultant.
Expression: AB BC AC Expression: OA OB OC

Mathematical Form of Law of Parallelogram:

Let forces P & Q act along OA and OB in both magnitude and direction respectively. Their resultant R acts
along OC. Angle between P & Q is α. θ is the angle between P & R. [β is the angle between Q & R]
Draw CD perpendicular to OA extended from C that intersects the extended OA at point D.
In figure, OA P P, OB AC Q Q, OC R R

According to the law of Parallelogram:

P Q R
Here, OB||AC and OD is the transversal
∴ ∠BOA ∠CAD α
AD
Then, in ∆ACD, cosα
AC
⇒ AD AC cosα
∴ AD Q cosα
CD
sinα ⇒ CD AC sinα
AC
∴ CD Q sinα

9
 Determination of resultant value:
OC OD CD ∵ ∆OCD is a right angled triangle]
⇒R OA AD CD P Q cosα Q sinα
P Q cos α 2PQ cosα Q sin α P Q cos α sin α 2PQ cosα
⇒ R P Q 2PQ cosα ∴ R P Q 2PQ cosα

Determination of the direction of the resultant:

Angle between P and R, ∠COD θ, ∆COD is a right angled triangle.
∴ tanθ
+

∴ tanθ ∴θ tan−
+ +

Note
The vector that makes (P) θ angle with R remains below and nothing remains in multiplication with it.

Similarly, it can be proved by drawing a perpendicular on the extension of OB from C.

P sinα
tanβ [β is the angle between Q and R ] ∴ β tan−
Q+P cosα +

Maximum value of the resultant:

R P2 Q2 2PQ cosα

R will be found when the value of cosα is maximum. cosα 1⇒ α 0°

In that case, R P Q 2PQ P Q
∴R P Q [when α 0°

Minimum value of the resultant:

R P Q 2PQ cosα
If the value of cosα is minimum, then the value of
resultant will be minimum. cosα 1
∴ α 180°
In that case, R P Q 2PQ 1 P Q = Q P P~Q
∴R P∼Q ∼ ⇒ Difference [Subtract smaller from larger] [when α 180°]
If P > Q then, R P Q and if P < Q then, R Q P

If °

R P Q 2PQ cos 90°

∴ R P Q
This can be proved by Pythagoras theorem. In the figure R P Q
Let's think back to our starting example.

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