GRADE 11 UNIT 2 Vectors quantity Introduction | Summaries Physics

Unit 2:- Vector Quantity

Grade 11 Physics Note page6

UNIT TWO

2. Vectors quantity

Introduction

All physical quantities are either scalar or vector quantities.

Scalar quantity

 It is physical quantity that can be described only by their magnitude.

Some properties of scalars:-

 They are represented by numerical value (any number ) with units.

 The product of two scalars is scalar.

 Addition and multiplication of scalar is commutative.

 Subtraction and division are not commutative.

Vectors quantity

 It is physical quantity that can be described both by their magnitude and direction

Some properties of vector quantities:-

 The sum or the difference of two vectors is a vector.

 Vector addition is commutative

 Vector subtraction is not commutative.

 The product of any scalar and vector is vector.

 The dot product of any vector and vector is scalar.

 The cross product of any vector and vector is vector

 The two vectors are equal , if they have equal magnitude and the same direction

Representation of vector

Vector can be represented graphically and analytically

Representation of vector graphically

 Vector is graphically represented straight line with direction.

i.e an arrow .

 The length of arrow represents the magnitude of vector.

 The head of arrow represents the direction of vector.

Representation of vector analytically

 Vectors are analytically written by putting an arrow above the symbol

 Vectors quantities are also represented by making boldface alphabet

2.1 Types of vector

In general vectors may be divided into three types.

 Proper vectors

 Axial vectors

 Inertial or pseudo vectors

Proper vectors

 They are non zero vectors .

Example :- Displacement ,force, momentum etc

Axial vectors

 The vectors which act along the axis of rotation are called axial vectors.

example :- angular velocity, torque, angular momentum, angular acceleration

Pseudo or inertial vectors

 The vectors used to make an inertial frame of reference into inertial frame of

reference are called pseudo or inertial vectors.

Vector may be further divided as:

Position vector :-

 It is a vector that starts from origin.

 It is a vector used to describe the position of a point in space.

 It is a vector that represents the position of a point with reference to a fixed point .

20m

Vector B is located 20m from vector A

Unit vector :-

 It is a vector of unit length.

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UNIT TWO

2. Vectors quantity Introduction All physical quantities are either scalar or vector quantities. Scalar quantity  It is physical quantity that can be described only by their magnitude. Some properties of scalars:-  They are represented by numerical value (any number ) with units.  The product of two scalars is scalar.  Addition and multiplication of scalar is commutative.  Subtraction and division are not commutative. Vectors quantity  It is physical quantity that can be described both by their magnitude and direction Some properties of vector quantities:-  The sum or the difference of two vectors is a vector.  Vector addition is commutative  Vector subtraction is not commutative.  The product of any scalar and vector is vector.  The dot product of any vector and vector is scalar.  The cross product of any vector and vector is vector  The two vectors are equal , if they have equal magnitude and the same direction Representation of vector Vector can be represented graphically and analytically Representation of vector graphically  Vector is graphically represented straight line with direction. i.e an arrow.  The length of arrow represents the magnitude of vector.  The head of arrow represents the direction of vector. Representation of vector analytically  Vectors are analytically written by putting an arrow above the symbol  Vectors quantities are also represented by making boldface alphabet 2.1 Types of vector In general vectors may be divided into three types.  Proper vectors  Axial vectors  Inertial or pseudo vectors Proper vectors  They are non zero vectors. Example :- Displacement ,force, momentum etc Axial vectors  The vectors which act along the axis of rotation are called axial vectors. example :- angular velocity, torque, angular momentum, angular acceleration Pseudo or inertial vectors  The vectors used to make an inertial frame of reference into inertial frame of reference are called pseudo or inertial vectors. Vector may be further divided as: Position vector :-  It is a vector that starts from origin.  It is a vector used to describe the position of a point in space.  It is a vector that represents the position of a point with reference to a fixed point. B 20m

A Vector B is located 20m from vector A Unit vector :-  It is a vector of unit length.

 It is a vector of magnitude one  The basic unit vectors in rectangular coordinates are Y j

X i unit vector i is 1 unit in x direction k unit vector j is 1 unit in y direction Z unit vector k is 1 unit in z direction Collinear vector :-  They are vectors that are parallel to each other and act along the same line.  They are limited to one dimension.  They can be in the same direction or opposite direction.  They are vectors in which the components act along the same line.

Coplanar vectors:-  They are vectors in which the components act in the same /common plane.  They are limited to two dimension.  They are Vectors which are lying in same plane.

2005. A vector that represents the position of an object in relation to another object is called A. Unit vectors B. Position vector C. Coplanar vectors D. Collinear vectors 2.2. Resolution of vectors  It is the method of finding two or more than two components vectors, which if added give the original vector. Resolving :- splitting one vector into two component vectors  The two components have the same effect as the original vector when combined.  The component vectors form the side of right -angle triangle  They make up opposite side and adjacent sides of the triangle. Example:- what is the a) magnitude and b) direction of the two coplanar vectors in Figure below

4m 3m a) D^2 = 32 + 42= 9+16= D = √25 = 5 m b) tan θ=opposite/adjacent=4/3=1.333... θ = tan –1 1.333 ... = 53º 2.3 Vector addiction and subtraction  Vector can be added graphically as well as analytically Addition of vector graphically  The first technique for vector addiction is drawing diagrams.  This technique can be applied to Collinear or coplanar vectors(B/C the diagram will be two dimensional only ) Scale diagrams  This technique is very simple  Select a scale for your own vector.  Draw them to scale, then connects the head of first vector with the tail of the next vector in a correct direction.

(iii) Vector addition is distributive, i.e., m (A + B) = m A + m B 2.4. Multiplication of vectors Multiplication of vector with scalar  Vectors can be multiplied by scalars.  Multiplication of vector with a scalar gives vector.  Multiplication of vector with a scalar changes the magnitude of vector.  Multiplication of vector with a scalar not changes the direction of vector.  Multiplication of vector with negative scalar reverse the direction of vector.

2005. Two non-zero vectors A and B are related by A = CB where c is a scalar. If the two vectors have opposite directions, then one of the following is true about c? A. c is a positive number B. c is a negative number C. c = 1 D. c = 0 Multiplication of vector with Vector  There are two ways of vector multiplication - The scalar /Dot/ product of vector - The Vector/ cross / product of vector. The scalar /Dot product of vector  It is denoted by. (dot).  The scalar product is also known as the dot product.  The scalar product of two vectors is scalar quantity.  The scalar product of two vectors is defined as a • b = axbx + ayby where the vectors are given in component form and are a = [ axay] and b = [bxby]. ax and bx are the components in the x-direction and ay and by are the components in the y-direction The scalar product can also be expressed as: a • b = |a| |b| cos θ where |a| and |b| are the magnitudes of the vectors a and b, respectively, and θ is the angle between the two vectors. we can calculate the angle between two vectors: cos θ = (a • b)/(|a| |b|) Properties of Scalar Product (i) Scalar product is commutative, i.e., A * B= B * A (ii) Scalar product is distributive, i.e., A * (B + C) = A * B + A * C (iii) Scalar product of two perpendicular vectors is zero. A * B = AB cos 90° = O (iv) Scalar product of two parallel vectors is equal to the product of their magnitudes, i.e., A * B = AB cos 0° = AB (v) Scalar product of a vector with itself is equal to the square of its magnitude i.e., A * A = AA cos 0° = A (vi) Scalar product of orthogonal unit vectors i. j = i. k = j. k = 0 and i. i = j. j = k. k = 1 (vii) Scalar product in Cartesian coordinates A. B = ( Axi + Ayj + Azk). (Bxi + Byj + Bzk ) = AxBx + AyBy + AzBz Vector or Cross Product of Two Vectors  It is denoted by * (cross).  The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. A * B = AB sin θ n  The direction of unit vector n can be obtained from right hand thumb rule.  If fingers of right hand are curled from A to B through smaller angle between them, then thumb will represent the direction of vector (A * B).  The vector or cross product of two vectors is also a vector. Properties of Vector Product (i) Vector product is not commutative, i.e., A * B ≠ B * A [∴ (A * B) = — (B * A)]

(ii) Vector product is distributive i.e., A * (B + C) = A * B + A * C (iii) Vector product of two parallel vectors is zero. i.e., A * B = AB sin O° = 0 (iv) Vector product of any vector with itself is zero. A * A = AA sin O° = 0 (v) Vector product of orthogonal unit vectors i X j = k i x k = j k x j = i and i. i = j. j = k. k = 0 (vi) Vector product in cartesian coordinates or If the vectors are in three dimensions A x B = (Axi + Ayj + Azk ) x ( Bxi + Byj + Bzk ) = i j k Ax Ay Az Bx By Bz = (AyBz-AzBy )i + ( AzBx - AxBz )j + (AyBx - AxBy )k Application of vectors Vectors have many applications. They are extremely useful in physics and many other areas. Some applications are as follows.

Analyzing forces on a bridge.
Analyzing the motion of an airplane.
Programming motion or the position of an object in a computer game or animation. •Displaying graphics (in the form of vector graphics) so that the diagram can be resized easily without any loss of quality.
Modeling and planning the trajectory (path) of a space probe.
Analyzing the motion of planets.
Analyzing magnetic fields. We can use vectors whenever there is a variable that has direction as well as magnitude. 2005. If A =8i+j-2k and B =5i-3j+k ,then, what is the value of B x A? A. 3i + 4j - 3k B. -3i - 4j + 3k C. -5i- 18j - 29k D. 5i + 18j + 29k 2006. Two non zero vectors D and E have precisely equal magnitudes. For the magnitude of D + E to be 3 times larger than the magnitude of D - E, what must be the angle between D and E? (A) 30° (B) 37° (C) 53° (D) 60° 2006. The vector 2i + j + k is perpendicular to i - 4j + λ k if λ is equal to (A) 0 (B) -1 (C) 2 (D) 3 2007. What is the angle between vectors A = (ai + a√3j ) units and B = (a√3 i + a j ) units? A. 90° B. 60° C. 37° D. 30° 2007. The magnitudes of two vectors A and B are 12 units and 5 units, respectively. Which one of the following is NOT the possible value for the magnitude of the resultant vector A+B? A. 60 units B. 17 units C. 13 units D. 7 units 2007. What is the vector product A x B of two vectors A = 7i + 4j - 8k and B = 3i - 2j + 5k? A. 36i + 11j - 2k B. 4i - 59j - 26k C. -4i + 59j + 26k D. -36i - 11j + 2k 2008. Determine the torque due to a force F = 2i - 3j + 4k N acting at a point r = 3i + 2j + 3k about the origin? A. 6i - 6j + 12k B. -6i + 6j + 12k C. 17i + 6j - 13k D. -17i + 6j + 13k

GRADE 11 UNIT 2 Vectors quantity Introduction, Summaries of Physics

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UNIT TWO