Vectors Class 11 Physics
Syllabus: Content to Study
- Definition and representation of vectors
- Types of vectors (unit vector, null vector, equal vectors)
- Vector addition and subtraction
- Resolution of vectors
- Vector multiplication (scalar and vector product)
- Applications of vectors in physics
Scalars and vectors:
Scalars are the physical quantities that can be represented by their magnitude. Examples of such physical quantities include mass, time, length, energy, temperature etc. All of these require a magnitude to be represented. For instance mass is represented by just expressing its magnitude in respective units, like 5 kg, time is expressed in seconds, energy in joules, temperature in Celsius. There is no sense of direction in these physical quantities.
Vectors are the physical quantities that require magnitude and also direction to be represented. Examples include force, velocity, acceleration, etc. All of these require a magnitude and a direction in which they act to be completely defined. For instance force needs to be defined in terms of magnitude in N and also the direction in which it acts; velocity is expressed in ms-1 and needs a direction in which the body is travelling.,
Unit Vector:
A vector having a magnitude equal to 1 and it is denoted by putting a hat (circumflex) on top of the vector symbol. Unit vector = ${\rm{\hat u}}$
A unit vector in the x-direction can be denoted by or ${\rm{\hat i}}$ similarly, unit vector in y-direction can be denoted by ${\rm{j}}$ or this
Null vector:
A vector having a zero magnitude and arbitrary direction is called as null vector. For example, the result of the sum of a vector and its negative vector is always a null vector. A+ (-A) =0
Equal vectors:
Two vectors suppose A and B are said to be equal if they have the same magnitude and same direction regardless of the position of their starting points.
Negative of a vector:
Vector equal in magnitude with the other vector but in the opposite direction. Such vector is called as negative of a vector. Suppose we have two vectors A and B both have the same magnitude but they are opposite in direction so they will be written as: A = -B
Parallelogram law of vectors:
Parallelogram law of vector states that “If the vectors acting
simultaneously at a point both in direction and magnitude represented by
the adjacent sides of the parallelogram drawn from the point, then the
resultant of the vectors both in magnitude and direction are represented
by the diagonal of the parallelogram passing through that point”.
Here,${\rm{\: }}\mathop\to\limits_{\rm{P}} $ and ${\rm{\: \:
}}\mathop\to\limits_{\rm{Q}} $ are represented by the sides of
parallelogram OBCA and the $\mathop\to\limits_{\rm{R}} $ is represented by
OC such that
${\rm{\: \: }}\mathop\to\limits_{\rm{R}} $ =${\rm{\: \:
\: }}\mathop\to\limits_{{\rm{\: P}}} $ + ${\rm{\: \:
}}\mathop\to\limits_{\rm{Q}} $
Let the angle between ${\rm{\: }}\mathop\to\limits_{\rm{P}} $ and ${\rm{\:
\: }}\mathop\to\limits_{\rm{Q}} $ is ${\rm{\: \: }}\theta $
Magnitude of$\:\:\mathop\to\limits_{\bf{R}} $
To calculate ${\rm{\: \: }}\mathop\to\limits_{\rm{R}} $ , draw the
perpendicular from point C to D as shown in figure.
In $\Delta $OCD
${\rm{O}}{{\rm{C}}^2}{\rm{\: }} = {\rm{\: \: }}{\left( {{\rm{OA}} +
{\rm{AD\: }}} \right)^2}{\rm{\: }} + {\rm{C}}{{\rm{D}}^2}$ ………………………i
In Δ ACD,
CD = Q sin θ
AD= Q cos θ
From I
${\rm{O}}{{\rm{C}}^2}{\rm{\: }} = {\rm{\: \: }}{\left( {{\rm{P}} +
{\rm{Qcos}}\theta {\rm{\: }}} \right)^2}{\rm{\: }} + {\left(
{{\rm{Qsin}}\theta {\rm{\: }}} \right)^2}$
${\rm{R\: }} = {\rm{\: }}\sqrt {{\rm{\: }}{{\rm{P}}^2} + {\rm{\:
}}2{\rm{PQcos}}\theta {\rm{\: }} + {\rm{\: \: }}{{\rm{Q}}^2}} $
Direction of ${\rm{\vec R}}$ is given by
Tan α $ = \frac{{{\rm{CD}}}}{{{\rm{OD}}}}{\rm{\: }}$
= $\frac{{{\rm{Q\: sin}}\theta }}{{\left( {{\rm{P\: }} + {\rm{\:
Q\:cos}}\theta {\rm{\: }}} \right){\rm{\: }}0}}{\rm{\: }}$
α = ${\tan ^{ - 1}}{\rm{\: Q}}\frac{{{\rm{sin}}\theta }}{{\left( {{\rm{P\:
}} + {\rm{\: Q\:cos}}\theta {\rm{\: }}} \right)}}$
Special cases,
-
when P and Q act on the same direction:θ = 0 °
${\rm{R\: \: }} = {\rm{\: }}\sqrt {{\rm{\: }}{{\rm{P}}^2} + {\rm{\: }}2{\rm{PQcos}}0{\rm{\: }} + {\rm{\: \: }}{{\rm{Q}}^2}} $
R = P+Q
Tan α = 0 -
When they act in opposite directionθ = 180 °
${\rm{R\: }} = {\rm{\: }}\sqrt {{\rm{\: }}{{\rm{P}}^2} + {\rm{\: }}2{\rm{PQcos}}180{\rm{\: }} + {\rm{\: \: }}{{\rm{Q}}^2}} $
R = A – B
Tan α = 0 -
When P an Q are perpendicular to each other: θ = 90
R = ${\rm{\: }}\sqrt {{{\rm{P}}^2} + {\rm{\: }}{{\rm{Q}}^2}} $
Tan α = $\frac{{\rm{Q}}}{{\rm{P}}}$
α = $\frac{{{{\tan }^1}{\rm{Q}}}}{{\rm{P}}}$
The triangle law of vectors states:
If two vectors such as AB and BC are representing the two sides of a
triangle ABC, then the third side AC closing the other side of the
triangle in opposite direction represents the sum of two vectors both in
magnitude and vectors.
$\overrightarrow {{\rm{AC}}} $= $\overrightarrow {{\rm{AB}}} $ +
$\overrightarrow {{\rm{BC}}} $
Rectangular components of vector:
Consider a vector ${\rm{\vec V\: }}$acting at a point making an angle θ
with positive X-axis as shown in figure Vector${\rm{\vec V}}$ is
represented by a line OA. From point A draw a perpendicular AB on X-axis.
Suppose OB and BA represents two vectors. Vector O’A is parallel to X-axis
and vector BA is parallel to Y-axis. Magnitude of these vectors are Vx and
Vy respectively. The sum of these vectors is equal to vector${\rm{\vec
V}}$.Thus Vx and Vy are the rectangular components of vector ${\rm{\vec
V\: }}$
Figure: from the figure,
Cosθ= $\frac{{{\rm{O\vec B}}}}{{{\rm{O}}\mathop\to\limits_{\rm{A}}{\rm{\:
}}}}$ =$\frac{{{{\rm{V}}_{\rm{x}}}}}{{\rm{V}}}$
${{\rm{V}}_{\rm{x}}}{\rm{\: }} = {\rm{Vcos\: }}\theta $
Sinθ=$\frac{{{\rm{A\vec B}}}}{{{\rm{O\vec A}}}}$=
$\frac{{{{\rm{V}}_{\rm{y}}}}}{{{\rm{V\: }}}}$
${{\rm{v}}_{\rm{y}}} = {\rm{V\: sin}}\theta $
Tanθ=$\frac{{{{\rm{v}}_{\rm{y}}}}}{{{{\rm{V}}_{\rm{x}}}}}$
The geometrical interpretation of the scalar product of two vectors:
If the product of two vectors is a scalar quantity, then the operation is
called scalar product.
Let us consider two vectors$\overrightarrow {{\rm{\: P}}} $and${\rm{\vec
Q}}$inclined at an angle θ to each other as shown in figure
Their scalar product is defined as${\rm{\: \vec P}}$${\rm{\: \vec Q}}$ . =
PQ cos θ
Work done by the force in moving through the displacement of D⃗ is scalar
product
W =${\rm{\: }}\overrightarrow {{\rm{F\: }}} $${\rm{\: \vec D}}$ = FD cos
θ
The scalar product of two vectors is a dot product.
Two vectors can be added by using the triangle law of vectors
If two vectors such as $\overrightarrow {{\rm{AB}}} $ and $\overrightarrow
{{\rm{BC}}} $ are representing the two sides of a triangle ABC, then the
third side AC closing the other side of the triangle in opposite direction
represents the sum of two vectors both in magnitude and vectors.
$\overrightarrow {{\rm{AC}}} $= $\overrightarrow {{\rm{AB}}} $ +
$\overrightarrow {{\rm{BC}}} $
By using parallelogram law of vectors
Statement:
"If two vectors are acting simultaneously at a point and these are
represented in magnitude and direction by any two adjacent side of a
triangle taken in order then the remaining side of the triangle gives the
magnitude and direction at their resultant taken in opposite sides.
Explanation:
Let ${\rm{\vec P\: and\: \vec Q}}$ be the two vectors acting
simultaneously at a point O and these are represented in magnitude and
direction by two adjacent sides AB and BC of a triangle ABC then the side
AC of a triangle ABC represent their resultant in magnitude and direction
as shown in figure
i.e.$\frac{{\overrightarrow {{\rm{AC}}} }}{{{\rm{\vec R}}}} =
\frac{{\overrightarrow {{\rm{AB}}} }}{{{\rm{\vec P}}}} +
\frac{{\overrightarrow {{\rm{BC}}} }}{{{\rm{\vec Q}}}}$
Addition of three vectors
For the addition of three vectors polygon law of vectors is used.
$\mathop\to\limits_{\rm{R}} = {\rm{\: }}\mathop\to\limits_{\rm{A}} +
{\rm{\: }}\mathop\to\limits_{\rm{B}} {\rm{\: }} + {\rm{\:
}}\mathop\to\limits_{\rm{C}} + {\rm{\: }}\mathop\to\limits_{\rm{D}}
{\rm{\: }}$
The resultant is the vector sum of two or more vectors. It is the result
of adding two or more vectors together. If displacement vectors A, B, C,D
,E are added together, the result will be vector R as shown in the
diagram, vector R can be determined.