General Physics Lecture notes Presented by

Lec.Dr. Sarah Mahdi Obaid

Department of Biomedical Engineering,

Al-Mustaqbal University College,

Babil, Iraq

Email: sarah.mahdi@mustaqbal-college.edu.iq

First year students

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

1

Vectors

1.1 Vectors and Scalars

Vectors are mathematical quantities that are useful to describe physical quantities that have both a magnitude

and direction associated with them.

Vector: any quantity that has a magnitude and direction and behaves like the displacement, e.g. displacement,

force, velocity, momentum, acceleration and torque.

Scalar: any quantity that has a magnitude but no direction e.g. length, density, time, mass, temperature and

energy.

1.2 Vector addition

To add two vector, 𝐴 and �⃗⃗�, draw vector 𝐴, then place the tail of �⃗⃗� at the head of vector 𝐴, then connect

the tail of 𝐴 to the head of �⃗⃗� to obtain the vector (sum or resultant) 𝐶 :

𝐴 �⃗⃗� 𝐴

�⃗⃗�

𝐶 = 𝐴 + �⃗⃗� �⃗⃗�

𝐴

𝐶

𝐶 = 𝐴 + �⃗⃗�

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

2

= 𝐴 + �⃗⃗� = �⃗⃗� + 𝐴 (commutative law)

= (𝐴 + �⃗⃗�) + 𝐶

= 𝐴 + (�⃗⃗� + 𝐴)

Let 𝐴 = 𝐴𝑥𝑖̂ + 𝐴𝑦𝑗̂ + 𝐴𝑧�̂� , �⃗⃗� = 𝐵𝑥𝑖̂ + 𝐵𝑦𝑗̂ + 𝐵𝑧�̂�

𝐴 + �⃗⃗� = (𝐴𝑥 + 𝐵𝑥)𝑖̂ + (𝐴𝑦 + 𝐵𝑦)𝑗̂ + (𝐴𝑧 + 𝐵𝑧)�̂�

𝐴 − �⃗⃗� = (𝐴𝑥 − 𝐵𝑥)𝑖̂ + (𝐴𝑦 − 𝐵𝑦)𝑗̂ + (𝐴𝑧 − 𝐵𝑧)�̂�

Subtraction: The subtraction of two vectors 𝐴 and �⃗⃗� is defined as the sum of 𝐴 and(−�⃗⃗�).

𝐶 = 𝐴 − �⃗⃗� = 𝐴 − �⃗⃗� = 𝐴 + (−�⃗⃗�)

𝐴

�⃗⃗�

𝐶 �⃗⃗�

𝐴

�⃗⃗�

𝐴

−�⃗⃗�

𝐶 = 𝐴 + �⃗⃗�

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

3

1.3 Vector Components

A vector is completely described by its components. Components can be used instead of vector itself.

Components of vector is the projection of the vector on an axis

𝑟 = 𝑟𝑥 + 𝑟𝑦

𝑟𝑥 = 𝑟𝑐𝑜𝑠𝜃

𝑟𝑦 = 𝑟𝑠𝑖𝑛𝜃

The process of finding the components of a vector is called

resolving the vector.

The magnitude direction of 𝑟 are written as:

𝑟 = √𝑟𝑥2 + 𝑟𝑦

2

𝑡𝑎𝑛𝜃 =𝑟𝑦

𝑟𝑥

𝑥

𝑦

𝜃

𝑟 𝑟𝑦⃗⃗⃗ ⃗

𝑟𝑥⃗⃗⃗ ⃗

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

4

1.4 Unit Vector:

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. We shell use the

symbols of 𝑖,̂ 𝑗̂ and �̂� to respect unit vector pointing in positive direction of x, y and z axis

For example, we can express 𝑟 of figure as

𝑟 = 𝑥𝑖̂ + 𝑦𝑗̂

In 3-dimension (3-D)

𝑟 = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧�̂�

Example (1): 2-Dimension

𝐴𝑥 = 𝐴𝑐𝑜𝑠𝜃𝑥

𝐴𝑦 = 𝐴𝑠𝑖𝑛𝜃𝑥

magnitude of 𝐴

|𝐴| = 𝐴 = √𝐴𝑥2 + 𝐴𝑦

2

𝐴 = 𝐴𝑥𝑖̂ + 𝐴𝑦𝑗̂ = 𝐴𝑐𝑜𝑠𝜃𝑥𝑖̂ + 𝐴𝑠𝑖𝑛𝜃𝑥𝑗̂

Example (2): General 3-D vector

𝑦

𝑥𝑖 ̂

𝑥

𝑦𝑗̂

𝜃

𝑥

𝑦

𝜃𝑥

𝜃𝑦 𝐴𝑦

𝐴𝑥

𝛼

𝛾

𝛽

𝑥

𝑦

𝑧

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

5

𝐴𝑥 = 𝐴𝑐𝑜𝑠𝛼

𝐴𝑦 = 𝐴𝑐𝑜𝑠𝛽

𝐴𝑧 = 𝐴𝑐𝑜𝑠𝛾

𝑐𝑜𝑠𝛼2 + 𝑐𝑜𝑠𝛽2 + 𝑐𝑜𝑠𝛾2 = 1

Magnitude of 𝐴 vector is written as |𝐴| = 𝐴, |𝐴| = √𝐴𝑥2 + 𝐴𝑦

2 + 𝐴𝑧2

𝑐𝑜𝑠𝛼 =𝐴𝑥

𝐴, 𝑒𝑡𝑐.

Let 𝑟 = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧�̂� and 𝐴 = 𝐴𝑐𝑜𝑠𝛼𝑖̂ + 𝐴𝑐𝑜𝑠𝛽𝑗̂ + 𝐴𝑐𝑜𝑠𝛾�̂�

1.5 Vector Multiplication

There are two ways to multiply vectors:

a. Dot product (scalar product): The scalar product of vectors 𝐴 and �⃗⃗� is written as 𝐴. �⃗⃗� and define to

be:

𝐴. �⃗⃗� = |𝐴||�⃗⃗�| cos 𝜃 = 𝐴𝐵 𝑐𝑜𝑠𝜃 𝜃 ≤ 180°

where |𝐴| is the magnitude of 𝐴.

|�⃗⃗�| is the magnitude of �⃗⃗�. 𝐴

�⃗⃗�

𝜃

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

6

𝜃 is the angle between 𝐴 and �⃗⃗�

Some properties of scalar product:

1. 𝐴. �⃗⃗� = �⃗⃗�. 𝐴 (commutative)

2. 𝐴. 𝐴 = |𝐴||𝐴|𝑐𝑜𝑠𝜃

= 𝐴2 (Parallel product)

3. 𝐴. (−𝐴) = −𝐴2 (anti parallel product)

4. 0 < 𝜃 < 90° (𝐴. �⃗⃗�) > 0

90° < 𝜃 < 180° (𝐴. �⃗⃗�) < 0

𝜃 = 90° (𝐴. �⃗⃗�) = 0

5. 𝑖.̂ 𝑖̂ = 𝑗̂. 𝑗̂ = �̂�. �̂� = 1 (parallel unit vector)

(𝑖.̂ 𝑗̂ = 𝑗̂. 𝑖̂ = 0) ,(𝑖.̂ �̂� = �̂�. 𝑖̂ = 0), (𝑗̂. �̂� = �̂�. 𝑗̂ = 0) (perpendicular unit vector)

6. (𝐶 + �⃗⃗⃗�). �⃗⃗� = 𝐶. �⃗⃗� + �⃗⃗⃗�. �⃗⃗� (distribution law)

𝐴. �⃗⃗� = (𝐴𝑥𝑖̂ + 𝐴𝑦𝑗̂ + 𝐴𝑧�̂�). (𝐵𝑥𝑖̂ + 𝐵𝑦𝑗̂ + 𝐵𝑧�̂�)

[(𝐴𝑥𝐵𝑥𝑖.̂ 𝑖̂ + 𝐴𝑥𝐵𝑦𝑖.̂ 𝑗̂ + 𝐴𝑥𝐵𝑧𝑖.̂ �̂�) + (𝐴𝑦𝐵𝑥𝑗̂. 𝑖̂ + 𝐴𝑦𝐵𝑦𝑗̂. 𝑗̂ + 𝐴𝑦𝐵𝑧𝑗̂. �̂�) + (𝐴𝑧𝐵𝑥�̂�. 𝑖̂ + 𝐴𝑧𝐵𝑦�̂�. 𝑗̂ + 𝐴𝑧𝐵𝑧�̂�. �̂�)]

𝐴

𝐴

𝐴

−𝐴

𝐴. �⃗⃗� = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵𝑧

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

7

What is 𝑗̂. �̂� = 𝑗̂. (𝐴𝑥𝑖̂ + 𝐴𝑦𝑗̂ + 𝐴𝑧�̂�)?

Solution: it’s equal to 𝐴𝑦

First application of a scalar product will be the concept of work which given by

𝑊 = �⃗�. 𝑑

Example 3: What is the right angle between 𝐴 = 3𝑖̂ + 7�̂� and �⃗⃗� = −𝑖̂ + 2𝑗̂ + �̂�?

Solution:

𝐴𝒙 = 3 , 𝐵𝒙 = −1

𝐴𝑦 = 0 , 𝐵𝒚 = 2

𝐴𝑧 = 7 , 𝐵𝑧 = 1

𝐴. �⃗⃗� = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵𝑧

= (3)(−1) + (0)(2) + (7)(1)

= +4

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

8

|𝐴| = √𝐴𝑥2 + 𝐴𝑦

2 + 𝐴𝑧2

= √32 + 02 + 72 ⟹ |𝐴| = √58

|�⃗⃗�| = √𝐵𝑥2 + 𝐵𝑦

2 + 𝐵𝑧2

= √(−1)2 + 22 + 12 ⟹ |�⃗⃗�| = √6

𝑐𝑜𝑠𝜃 =�⃗�.�⃗⃗�

|�⃗�||�⃗⃗�|⟹ 𝑐𝑜𝑠𝜃 =

4

√58 √6

𝜃 = 𝑐𝑜𝑠−1 4

√58 √6

𝜃 = 77.6° [The angle between two vector 𝐴 and �⃗⃗� ]

b. Cross product (vector product):

The vector product of vectors 𝐴 and �⃗⃗� is written as 𝐴 × �⃗⃗� and define to be:

𝐴 × �⃗⃗� = |𝐴||�⃗⃗�|𝑠𝑖𝑛𝜃�̂� , where �̂� =�⃗⃗�

|𝑛|

Where the vector product produced another vector perpendicular the plane that contain 𝐴 and �⃗⃗�.

𝐶 ⊥ to both 𝐴 and �⃗⃗�.

𝐴 �⃗⃗�

𝐶

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

9

Some properties of vector product:

1. 𝐴 × �⃗⃗� = −�⃗⃗� × 𝐴

2. 𝐴 × 𝐴 = |𝐴||𝐴|𝑠𝑖𝑛0 = 0 [parallel vector]

3. (𝑖̂ × 𝑖)̂ = (𝑗̂ × 𝑗̂) = (�̂� × �̂�) = 0 [parallel unit vector]

4. {

𝑖̂ × 𝑗̂ = −𝑗̂ × 𝑖̂ = �̂�

�̂� × 𝑖̂ = −𝑖̂ × �̂� = 𝑗̂

𝑗̂ × �̂� = −�̂� × 𝑗̂ = 𝑖̂

} 𝑅𝑖𝑔ℎ𝑡 ℎ𝑎𝑛𝑑 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚

𝐴 × �⃗⃗� = (𝐴𝑥𝑖̂ + 𝐴𝑦𝑗̂ + 𝐴𝑧�̂�) × (𝐵𝑥𝑖̂ + 𝐵𝑦𝑗̂ + 𝐵𝑧�̂�)

= (𝐴𝑦𝐵𝑧 − 𝐴𝑧𝐵𝑦)𝑖̂ − (𝐴𝑧𝐵𝑥 − 𝐴𝑥𝐵𝑧)𝑗̂ + (𝐴𝑥𝐵𝑦 − 𝐴𝑦𝐵𝑥)�̂� [Result following multiplication term-by

term and dimension of many vanishing terms]

𝐴 × �⃗⃗� = |

𝑖̂ 𝑗̂ �̂�𝐴𝑥 𝐴𝑦 𝐴𝑧

𝐵𝑥 𝐵𝑦 𝐵𝑧

|

1.6 Determinates

a. Order-2

|𝑎1 𝑎2

𝑏1 𝑏2| = 𝑎1𝑏2 − 𝑎2𝑏1

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

10

Example (4): |3 −24 5

| = (3)(5) − (−2)(4) = 23

b. Order-3

|

𝑎1 𝑎2 𝑎3

𝑏1 𝑏2 𝑏3

𝑐1 𝑐2 𝑐3

| = 𝑎1 |𝑏2 𝑏3

𝑐2 𝑐3| − 𝑎2 |

𝑏1 𝑏3

𝑐1 𝑐3| + 𝑎3 |

𝑏1 𝑏2

𝑐1 𝑐2|

Example (5): |3 2 −14 3 3

−2 7 1| = 3 |

3 37 1

| − 2 |4 3

−2 1| + (−1) |

4 3−2 7

|

= 3[3(1) − 3(7)] − 2[4(1) − 3(−2)] + (−1)[4(7) − 3(−2)] = −108

Example (6): if 𝐴 = 3𝑖̂ + 7𝑗̂ − �̂� , �⃗⃗� = 𝑖̂ − 𝑗̂ what is 𝐶 = 𝐴 × �⃗⃗�, 𝐶. 𝐴 and 𝐶. �⃗⃗�?

Solution: 𝐴 × �⃗⃗� = |𝑖̂ 𝑗̂ �̂�3 7 −11 −1 0

|

= 𝑖[̂7(0) − (−1)(−1)] − 𝑗̂[3(0) − (−1)(1)] + �̂�[3(−1) − 7(1)]

= −𝑖̂ − 𝑗̂ + 10�̂�

𝐶. 𝐴 = (−𝑖̂ − 𝑗̂ + 10�̂�). (3𝑖̂ + 7𝑗̂ − �̂�) = −3 − 7 + 10 = 0

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

11

𝐶. �⃗⃗� = (−𝑖̂ − 𝑗̂ + 10�̂�). (𝑖̂ − 𝑗̂) = −1 + 1 = 0

𝐶 ⊥ to both 𝐴 and �⃗⃗�

Example (7): find the magnitude and direction of the sum of three vector 𝐴, �⃗⃗� and 𝐶 lying in the xy plane

and given by:

𝐴 = 4.2𝑖̂ − 1.5𝑗̂

�⃗⃗� = −1.6𝑖̂ + 2.9𝑗̂

𝐶 = −3.7𝑗̂

Solution:

𝑅𝑥 = 𝐴𝑥 + 𝐵𝑥 + 𝐶𝑥

4.2 − 1.6 + 0 = 2.6

𝑅𝑦 = 𝐴𝑦 + 𝐵𝑦 + 𝐶𝑦

= −1.5 + 2.9 − 3.7 = −2.3

�⃗⃗� = 𝑅𝑥𝑖̂ + 𝑅𝑦𝑗̂

2.6𝑖̂ − 2.3𝑗̂

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

12

The magnitude of �⃗⃗�:

𝑅 = √𝑅𝑥2 + 𝑅𝑦

2

= √(2.6)2 + (−2.3)2

𝑅 = 3.5

The direction of �⃗⃗�:

𝑡𝑎𝑛∅ =𝑅𝑦

𝑅𝑥⟹ ∅ = −41

1.7 Differentiation of Vector:

Let 𝐴(𝑡) is a vector as a function to the variable t, where:

𝐴(𝑡) = 𝑖̂𝐴𝑥(𝑡) + 𝑗̂𝐴𝑦(𝑡) + �̂�𝐴𝑧(𝑡)

Then the differentiation process due to t is:

𝑑�⃗�

𝑑𝑡= 𝑖̂

𝑑𝐴𝑥

𝑑𝑡+ 𝑗̂

𝑑𝐴𝑦

𝑑𝑡+ �̂�

𝑑𝐴𝑧

𝑑𝑡

For two vector 𝐴 and �⃗⃗�, we can write:

Department of Biomedical Engineering Lecture notes by Dr. Sarah Mahdi Obaid Al-Mustaqbal University College

13

𝑑

𝑑𝑡(𝐴 + �⃗⃗�) =

𝑑�⃗�

𝑑𝑡+

𝑑�⃗⃗�

𝑑𝑡

𝑑

𝑑𝑡(𝐴. �⃗⃗�) = �⃗⃗�.

𝑑�⃗�

𝑑𝑡+ 𝐴.

𝑑�⃗⃗�

𝑑𝑡