By: Engr. Hinesh Kumar Lecturer I.B.T, LUMHS, Jamshoro SCALAR AND VECTOR By: Engr. Hinesh Kumar Lecturer I.B.T, LUMHS, Jamshoro

Scalars Scalars are quantities which have magnitude without direction. Examples of scalars time amount density charge temperature mass kinetic energy

Vector A vector is a quantity that has both magnitude (size) and direction. it is represented by an arrow whereby the length of the arrow is the magnitude, and the arrow itself indicates the direction

Contd…. A The symbol for a vector is a letter with an arrow over it. All vectors have head and tail. A

Two ways to specify a vector x  A It is either given by a magnitude A, and a direction  Or it is given in the x and y components as Ax Ay A y x Ax Ay

Ax = A cos  Ay = A sin  │A │ =√ ( Ax2 + Ay2 ) The magnitude (length) of A is found by using the Pythagorean Theorem │A │ =√ ( Ax2 + Ay2 ) The length of a vector clearly does not depend on its direction.

tan  = Ay / Ax  =tan-1(Ay / Ax) The direction of A can be stated as

Vector Representation of Force Force has both magnitude and direction and therefore can be represented as a vector.

Vector Representation of Force The figure on the left shows 2 forces in the same direction therefore the forces add. The figure on the right shows the man pulling in the opposite direction as the cart and forces are subtracted.

Some Properties of Vectors Equality of Two Vectors Two vectors A and B may be defined to be equal if they have the same magnitude and point in the same directions. i.e. A = B A A B B A B

Negative of a Vector The negative of vector A is defined as giving the vector sum of zero value when added to A . That is, A + (- A) = 0. The vector A and –A have the same magnitude but are in opposite directions. A -A

Applications of Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? + 54.5 m, E 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION. 84.5 m, E

C = A + B Vector Addition C B A Example The addition of two vectors A and B - will result in a third vector C called the resultant C = A + B Geometrically (triangle method of addition) put the tail-end of B at the top-end of A C connects the tail-end of A to the top-end of B A B C We can arrange the vectors as we like, as long as we maintain their length and direction Example

More than two vectors? Example x4 x5 xi x3 x2 xi = x1 + x2 + x3 + x4 + x5 x1 Example

Applications of Vectors VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E - 30 m, W 24.5 m, E

Vector Subtraction Example A B C = A + (-B) Equivalent to adding the negative vector A B C = A + (-B) A -B A - B B Example

Scalar Multiplication The multiplication of a vector A by a scalar  - will result in a vector B B =  A - whereby the magnitude is changed but not the direction Do flip the direction if  is negative

B =  A (A) = A = (A) (+)A = A + A If  = 0, therefore B =  A = 0, which is also known as a zero vector (A) = A = (A) (+)A = A + A Example

Rules of Vector Addition commutative A + B = B + A A B A + B

(A + B) + C = A + (B + C) associative B C A A + B (A + B) + C

distributive m(A + B) = mA + mB A B A + B mA mB m(A + B)

Parallelogram method of addition (tailtotail) A + B A B The magnitude of the resultant depends on the relative directions of the vectors

Unit Vectors and defined as a vector whose magnitude is 1 and dimensionless the magnitude of each unit vector equals a unity; that is, │ │= │ │= │ │= 1 i  j  k  and defined as  i a unit vector pointing in the x direction j a unit vector pointing in the y direction k a unit vector pointing in the z direction  

Useful examples for the Cartesian unit vectors [ i, j, k ] - they point in the direction of the x, y and z axes respectively x y z i j k

Component of a Vector in 2-D vector A can be resolved into two components Ax and Ay x- axis y- axis Ay Ax A θ A = Ax + Ay

│Ax│ = Ax = A cos θ │Ay│ = Ay = A sin θ The magnitude of A The component of A are x- axis y- axis Ay Ax A θ │Ax│ = Ax = A cos θ │Ay│ = Ay = A sin θ The magnitude of A A = √Ax2 + Ay2 The direction of A tan  = Ay / Ax  =tan-1(Ay / Ax)

A = Axi + Ayj j i The unit vector notation for the vector A is written y- axis Ax Ay θ A i j x- axis

Component of a Unit Vector in 3-D vector A can be resolved into three components Ax , Ay and Az A Ax Ay Az y- axis x- axis z- axis i j k A = Axi + Ayj + Azk

A + B = C sum of the vectors A and B can then be obtained as vector C if A = Axi + Ayj + Azk B = Bxi + Byj + Bzk A + B = C sum of the vectors A and B can then be obtained as vector C C = (Axi + Ayj + Azk) + (Bxi + Byj + Bzk) C = (Ax + Bx)i+ (Ay + By)j + (Az + Bz)k C = Cxi + Cyj + Czk

Dot product (scalar) of two vectors The definition: θ B A A · B = │A││B │cos θ

Dot product (scalar product) properties: if θ = 900 (normal vectors) then the dot product is zero |A · B| = AB cos 90 = 0 and i · j = j · k = i · k = 0 if θ = 00 (parallel vectors) it gets its maximum value of 1 |A · B| = AB cos 0 = 1 and i · j = j · k = i · k = 1

A · B = (Axi + Ayj + Azk) · (Bxi + Byj + Bzk) the dot product is commutative A + B = B + A Use the distributive law to evaluate the dot product if the components are known A · B = (Axi + Ayj + Azk) · (Bxi + Byj + Bzk) A. B = (AxBx) i.i + (AyBy) j.j + (AzBz) k.k A .

Cross product (vector) of two vectors The magnitude of the cross product given by θ A B C │C │= │A x B│ = │A││B │sin θ the vector product creates a new vector this vector is normal to the plane defined by the original vectors and its direction is found by using the right hand rule

Cross product (vector product) properties: if θ = 00 (parallel vectors) then the cross product is zero and i x i = j x j = k x k = 0 |A x B| = AB sin 0 = 0 if θ = 900 (normal vectors) it gets its maximum value |A x B| = AB sin 90 = 1 and i x i = j x j = k x k = 1

the relationship between vectors i , j and k can be described as i x j = - j x i = k j x k = - k x j = i k x i = - i x k = j fsddddadd

Example 1 (2 Dimension) If the magnitude of vector A and B are equal to 2 cm and 3 cm respectively , determine the magnitude and direction of the resultant vector, C for B A A + B 2A + B

Solution |A + B| = √A2 + B2 = √22 + 32 = 3.6 cm The vector direction tan θ = B / A θ = 56.3 |2A + B| = √(2A)2 + B2 = √42 + 32 = 5.0 cm The vector direction tan θ = B / 2A θ = 36.9

Example 2 Find the sum of two vectors A and B lying in the xy plane and given by A = 2.0i + 2.0j and B = 2.0i – 4.0j

Solution Comparing the above expression for A with the general relation A = Axi + Ayj , we see that Ax= 2.0 and Ay= 2.0. Likewise, Bx= 2.0, and By= -4.0 Therefore, the resultant vector C is obtained by using Equation C = A + B + (2.0 + 2.0)i + (2.0 - 4.0)j = 4.0i -2.0j or Cx = 4.0 Cy = -2.0 The magnitude of C given by equation Tan θ = Ry / Rx = (Ax + By) / (Ax + Bx) C = √Cx2 + Cy2 = √20 = 4.5 Exercise Find the angle θ that C makes with the positive x axis

Example - 2D [headtotail] (2, 2) (1, 0)

Solution x1 + x2 = (1, 0) + (2, 2) = (3, 2) x1 + x2 x1 x2

Example - 2D [tailtotail] (2, 2) (1, 0)

Solution x1 - x2? (x2) x1 x1 + x2 x2 x1 + x2 = (1, 0) + (2, 2) = (3, 2) x1 - x2?

Example of 2D (Subtraction) (2, 2) (1, 0)

Solution x1 - x2 = x1 + (-x2) x1 - x2 = (1, 0) - (2, 2) = (-1, -2) x1

Example -2D for subtraction (2, 2) (1, 0)

Assignment If one component of a vector is not zero, can its magnitude be zero? Explain and Prove it. 1 If A + B = 0, what can you say about the components of the two vectors? 2 A particle undergoes three consecutive displacements d1 = (1.5i + 3.0j – 1.2k) cm, d2 = (2.3i – 1.4j – 3.6k) cm d3 = (-1.3i + 1.5j) cm. Find the component and its magnitude. 3