GED0081 – College Physics 1
Multiplication
of Vectors
MPS Department | FEU Institute of Technology | 2.2
Dot Product
Objectives
At the end of the lecture, the students must be able to:
• Define dot product and cross product.
• Perform multiplication of vectors analytically and geometrically.
• Determine the angle between two vectors using dot product definition.
• Familiarize the right-hand rule.
There are two types of vector multiplication:
Dot Product and Cross Product
• Also called “scalar” product.
• Type of vector multiplication.
• Has TWO FORMULAS
• THE RESULT OF A DOT PRODUCT IS
ALWAYS A SCALAR QUANTITY
If the given are the magnitude of the vectors and
the angle between them, we use:
WHAT IF INSTEAD OF MAGNITUDES, THE COMPONENTS
ARE GIVEN? WHAT ARE WE GONNA DO NOW!?
𝑨 ∙ 𝑩 = 𝑨 𝑩 𝒄𝒐𝒔𝜽𝑨𝑩 = 𝑨𝒙 𝑩𝒙 + 𝑨𝒚 𝑩𝒚 + 𝑨𝒛 𝑩𝒁
// The first formula is used when the
magnitudes and angle are given.
// The second formula is used when the
vectors are given in their component
forms.
// Both formulas are usually used when
finding the angle between two vectors.
Cross Product
• Also called “vector” product.
• Type of vector multiplication.
• Has TWO FORMULAS
If the given are the magnitude of the vectors and
the angle between them, we use:
𝑨 × 𝑩 = 𝑨 𝑩 𝒔𝒊𝒏𝜽𝑨𝑩 https://www.physicsclassroom.com/class/vectors/Lesson-3/Cross-
IF YOU DO NOT WANT TO MEMORIZE THIS LENGTHY FORMULA, APPLY THE HASH
SLINGING SLASHER METHOD!
https://www.youtube.com/watch?v=2wTUqZa66ng
GED0081 – College Physics 1
Addition of
Vectors
MPS Department | FEU Institute of Technology | 2.1
Objectives
At the end of the lecture, the students must be able to:
• Differentiate resultant vector from equilibrant vector.
• Add two given vectors in 2D space.
• Add two given vectors in 3D space.
• Explain the meaning of “vector subtraction” using vector addition
• Add three or more vectors given in component form.
• The resultant is the vector sum of two or more vectors.
It is the result of adding two or more vectors together.
• Equilibrant is a vector that points on the opposite to
where the resultant is pointing to.
• Equilibrant and resultant have equal magnitudes but
opposite directions. https://www.quora.com/What-is-the-relation-between-a-resultant-force-and-an-
equilibrant-force-How-will-you-calculate-the-magnitude-of-a-resultant-of-two-force-
60N-and-40N-acting-at-a-point-such-that-they-are-perpendicular-to-each-other
IF THE RESULTANT VECTOR IS 5 METERS POINTING
NORTH-EAST, THEN THE EQUILIBRANT VECTOR IS 5
METERS POINTING SOUTH WEST.
In 1D, we add the second vector at the end of the first vector.
The vector from the tail of the first vector (the starting point) to the head of the second vector
(the end point) is then the sum of the vectors.
https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition
In 2D, draw the vectors and apply trigonometry or apply component method!
In 3D, apply component method!
A mountain climbing expedition establishes a base camp and two intermediate camps, A
and B. Camp A is 11,200 m east of and 3,200 m above base camp. Camp B is 8400 m east
of and 1700 m higher than Camp A. Determine the displacement between base camp and
Camp B.
https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition
LET’S SAY WE WANT TO ADD ARBITRARY VECTORS
𝑨 𝒂𝒏𝒅 𝑩, FOLLOW THESE STEPS:
Step 1: Draw the vectors in a coordinate plane.
Step 2: Identify the magnitudes and corresponding angles
with respect to the positive x-axis.
Step 3: Get the x and y-components of the first vector using
𝐴𝑥 = 𝐴Ԧ 𝑐𝑜𝑠𝜃𝐴 and 𝐴𝑦 = 𝐴Ԧ 𝑠𝑖𝑛𝜃𝐴
Step 4: Get x and y-components of the second vector using
𝐵𝑥 = 𝐵 𝑐𝑜𝑠𝜃𝐵 and 𝐵𝑦 = 𝐵 𝑠𝑖𝑛𝜃𝐵
Step 5: Add all the x-components, add all the y-components.
Step 6: Write your vector sum CONFIDENTLY WHILE SMILING.
GED0081 – College Physics 1
Module 2: Operations
on Vectors
MPS Department | FEU Institute of Technology
Subtopic 1
Addition of Vectors
Objectives
At the end of the lecture, the students must be able to:
• Differentiate resultant vector and equilibrant vector.
• Identify the different methods of adding vectors.
• Perform addition of two or more vectors.
• The resultant is the vector sum of two or more vectors.
It is the result of adding two or more vectors together.
• Equilibrant is a vector that points on the opposite to
where the resultant is pointing to.
• Equilibrant and resultant have equal magnitudes but
opposite directions. https://www.quora.com/What-is-the-relation-between-a-resultant-force-and-an-
equilibrant-force-How-will-you-calculate-the-magnitude-of-a-resultant-of-two-force-
60N-and-40N-acting-at-a-point-such-that-they-are-perpendicular-to-each-other
IF THE RESULTANT VECTOR IS 5 METERS POINTING
NORTH-EAST, THEN THE EQUILIBRANT VECTOR IS 5
METERS POINTING SOUTH WEST.
In 1D, we add the second vector at the end of the first vector.
The vector from the tail of the first vector (the starting point) to the head of the second vector
(the end point) is then the sum of the vectors.
https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition
In 2D, draw the vectors and apply trigonometry or apply component method!
In 3D, apply component method!
A mountain climbing expedition establishes a base camp and two intermediate camps, A
and B. Camp A is 11,200 m east of and 3,200 m above base camp. Camp B is 8400 m east
of and 1700 m higher than Camp A. Determine the displacement between base camp and
Camp B.
https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition
LET’S SAY WE WANT TO ADD ARBITRARY VECTORS
𝑨 𝒂𝒏𝒅 𝑩, FOLLOW THESE STEPS:
Step 1: Draw the vectors in a coordinate plane.
Step 2: Identify the magnitudes and corresponding angles
with respect to the positive x-axis.
Step 3: Get the x and y-components of the first vector using
𝐴𝑥 = 𝐴Ԧ 𝑐𝑜𝑠𝜃𝐴 and 𝐴𝑦 = 𝐴Ԧ 𝑠𝑖𝑛𝜃𝐴
Step 4: Get x and y-components of the second vector using
𝐵𝑥 = 𝐵 𝑐𝑜𝑠𝜃𝐵 and 𝐵𝑦 = 𝐵 𝑠𝑖𝑛𝜃𝐵
Step 5: Add all the x-components, add all the y-components.
Step 6: Write your vector sum CONFIDENTLY WHILE SMILING.
Subtopic 2
Dot Product
Objectives
At the end of the lecture, the students must be able to:
• Understand the physical meaning of dot product.
• Perform dot product on two vectors.
• Determine the angle between two vectors using dot product definition.
There are two types of vector multiplication:
Dot Product and Cross Product
• Also called “scalar” product.
• Type of vector multiplication.
• Has TWO FORMULAS
• THE RESULT OF A DOT PRODUCT IS
ALWAYS A SCALAR QUANTITY
If the given are the magnitude of the vectors and
the angle between them, we use:
WHAT IF INSTEAD OF MAGNITUDES, THE COMPONENTS
ARE GIVEN? WHAT ARE WE GONNA DO NOW!?
𝑨 ∙ 𝑩 = 𝑨 𝑩 𝒄𝒐𝒔𝜽𝑨𝑩 = 𝑨𝒙 𝑩𝒙 + 𝑨𝒚 𝑩𝒚 + 𝑨𝒛 𝑩𝒁
// The first formula is used when the
magnitudes and angle are given.
// The second formula is used when the
vectors are given in their component
forms.
// Both formulas are usually used when
finding the angle between two vectors.
Subtopic 3
Cross Product
Objectives
At the end of the lecture, the students must be able to:
• Understand the physical meaning of cross product.
• Perform cross product on two vectors.
• Apply right-hand rule in performing vector multiplication.
• Also called “vector” product.
• Type of vector multiplication.
• Has TWO FORMULAS
If the given are the magnitude of the vectors and
the angle between them, we use:
𝑨 × 𝑩 = 𝑨 𝑩 𝒔𝒊𝒏𝜽𝑨𝑩 https://www.physicsclassroom.com/class/vectors/Lesson-3/Cross-
IF YOU DO NOT WANT TO MEMORIZE THIS LENGTHY FORMULA, APPLY THE HASH
SLINGING SLASHER METHOD!
https://www.youtube.com/watch?v=2wTUqZa66ng
GED0081 – College Physics 1
Unit Vectors
MPS Department | FEU Institute of Technology | 1.2
Objectives
At the end of the lecture, the students must be able to:
• Define unit vectors.
• Identify the different components of a vector in 2D and 3D space.
• Calculate for unit vector representation of any given vector.
A vector which has a magnitude of 1.
= (0,0,1) which are of length 1 and
The basic unit vectors are 𝒊Ƹ = (1,0,0), 𝒋Ƹ = (0,1,0), and 𝒌
have directions along the positive x-axis, y-axis, and z-axis respectively.
// Every vector has
its own unit
vector
representation
described by the
image on the left.
http://mathonline.wikidot.com/standard-unit-vectors
http://mathonline.wikidot.com/standard-unit-vectors
A vector “A” can be represented by the notation: 𝑨
A vector “B” can be represented by the notation: 𝑩
The magnitude of 𝐴Ԧ can be represented by the notation: 𝑨
A vector 𝑨 in a 2-dimensional space is represented by:
𝑨 = 𝑨𝒙 𝒊Ƹ + 𝑨𝒚 𝒋Ƹ = 𝑨𝒙 , 𝑨𝒚 // We use the unit
vectors to
A vector 𝑨 in a 3-dimensional space is represented by: represent
= 𝑨𝒙 , 𝑨𝒚 , 𝑨𝒛
𝑨 = 𝑨𝒙 𝒊Ƹ + 𝑨𝒚 𝒋Ƹ + 𝑨𝒛 𝒌 components.
Example:
𝑨 = 𝟑𝒊Ƹ + 𝟐𝒋Ƹ = 𝟑, 𝟐
= 𝟔, −𝟏𝟎, 𝟏
𝑩 = 𝟔𝒊Ƹ − 𝟏𝟎𝒋Ƹ + 𝒌
𝑪 = 𝟏𝟎𝒋Ƹ + 𝒌 = 𝟎, −𝟏𝟎, 𝟏
To find a unit vector, 𝒖, in the same direction of a vector, 𝑨 , we use the formula:
𝑨 // Simply get the ratio of the
𝒖= vector itself and its magnitude.
𝑨
Example: Find the unit vector in the same direction of the vector 𝐹Ԧ = < 3, −4 >
Example: Find the unit vector in the same direction of the vector 𝑉 = 𝑖Ƹ + 2𝑗Ƹ − 𝑘
How many components does the vector <1,3,5> have?
How many components does the vector <0,-2,-6> have?
How many components does the vector 5i + 2j + 3k have?
How many components does the vector 3i - 5j - 3i have?
On what plane is the vector <1,0,-5> located?
On what plane is the vector <0,-7,-13> located?
GED0081 – College Physics 1
Vector
Representation
MPS Department | FEU Institute of Technology | 1.1
Objectives
At the end of the lecture, the students must be able to:
• Differentiate scalar and vector quantities.
• Identify the different representation of vectors
• Calculate for the magnitude of vectors in 2D and 3D space.
• Calculate for the direction of vectors in 2D space.
Lecture 1.1
Scalar and Vector
Quantities
“Some physical quantities, such as time, temperature, mass, and density, can be
described completely by a single number with a unit. But many other important
quantities in physics have a direction associated with them and cannot be described
by a single number.”
• When a physical quantity is described by a
single number, we call it a scalar quantity.
• In contrast, a vector quantity has both
magnitude and direction.
Vectors can be represented using arrows or components.
Vectors as arrows:
• A vector is drawn as an arrow with a head and a tail.
• The magnitude of the vector is often described by the length of the arrow.
• The arrow points in the direction of the vector.
https://www.grc.nasa.gov/www/k-12/airplane/vectcomp.html https://www.grc.nasa.gov/www/k-12/airplane/vectcomp.html
If two vectors have the same magnitude, they are equivalent.
If two vectors have the same direction, they are parallel.
If two vectors are pointing on opposite directions, they are called anti-parallel.
If they have the same magnitude and the same direction, they are equal.
https://www.grc.nasa.gov/www/k-12/airplane/vectcomp.html
For a vector 𝐴Ԧ in a 2-dimensional space, the magnitude can be calculated as:
𝑨 = 𝑨𝟐𝒙 + 𝑨𝟐𝒚
For a vector 𝐴Ԧ in a 3-dimensional space, the magnitude can be calculated as:
𝑨 = 𝑨𝟐𝒙 + 𝑨𝟐𝒚 + 𝑨𝟐𝒛
// Just think of it this way: to get the magnitude of a vector,
just get the square root of the sum of the squares of the
components ☺
For a vector 𝐴Ԧ in a 2-dimensional space, the magnitude can be calculated as:
−𝟏
𝑨𝒚
𝜽 = 𝐭𝐚𝐧
𝑨𝒙
Try these! TAKE NOTE!
Find the angle of inclination of the following vectors: The angle theta is actually the
1. <1.13, -3.22> angle of inclination of a vector with
respect to the positive x-axis. For a
2. <15.02, 29.16>
vector in 3D space, the angle theta
3. 4i + 7j does not completely describe its
4. <5,0> direction.
Find the angle of inclination of the following vectors:
1. <1.13, -3.22>
2. <15.02, 29.16>
3. 4i + 7j
4. <5,0>
GED0081 – College Physics 1
Module 1:
Vectors
MPS Department | FEU Institute of Technology
Subtopic 1
Introduction to
Vectors
“Some physical quantities, such as time, temperature, mass, and density, can be
described completely by a single number with a unit. But many other important
quantities in physics have a direction associated with them and cannot be described
by a single number.”
Objectives
At the end of the lecture, the students must be able to:
• Differentiate scalar and vector quantities.
• Define unit vectors
• Calculate for the magnitude and direction of vectors in 2D and 3D space.
• When a physical quantity is described by a
single number, we call it a scalar quantity.
• In contrast, a vector quantity has both
magnitude and direction.
Vectors can be represented using arrows or components.
Vectors as arrows:
• A vector is drawn as an arrow with a head and a tail.
• The magnitude of the vector is often described by the length of the arrow.
• The arrow points in the direction of the vector.
https://www.grc.nasa.gov/www/k-12/airplane/vectcomp.html https://www.grc.nasa.gov/www/k-12/airplane/vectcomp.html
If two vectors have the same magnitude, they are equivalent.
If two vectors have the same direction, they are parallel.
If two vectors are pointing on opposite directions, they are called anti-parallel.
If they have the same magnitude and the same direction, they are equal.
https://www.grc.nasa.gov/www/k-12/airplane/vectcomp.html
Subtopic 2
Unit Vectors
“Some physical quantities, such as time, temperature, mass, and density, can be
described completely by a single number with a unit. But many other important
quantities in physics have a direction associated with them and cannot be described
by a single number.”
Objectives
At the end of the lecture, the students must be able to:
• Define unit vectors.
• Identify the different components of a vector in 2D and 3D space.
• Calculate for unit vector representation of any given vector.
A vector which has a magnitude of 1.
= (0,0,1) which are of length 1 and
The basic unit vectors are 𝒊Ƹ = (1,0,0), 𝒋Ƹ = (0,1,0), and 𝒌
have directions along the positive x-axis, y-axis, and z-axis respectively.
// Every vector has
its own unit
vector
representation
described by the
image on the left.
http://mathonline.wikidot.com/standard-unit-vectors
http://mathonline.wikidot.com/standard-unit-vectors
A vector “A” can be represented by the notation: 𝑨
A vector “B” can be represented by the notation: 𝑩
The magnitude of 𝐴Ԧ can be represented by the notation: 𝑨
A vector 𝑨 in a 2-dimensional space is represented by:
𝑨 = 𝑨𝒙 𝒊Ƹ + 𝑨𝒚 𝒋Ƹ = 𝑨𝒙 , 𝑨𝒚 // We use the unit
vectors to
A vector 𝑨 in a 3-dimensional space is represented by: represent
= 𝑨𝒙 , 𝑨𝒚 , 𝑨𝒛
𝑨 = 𝑨𝒙 𝒊Ƹ + 𝑨𝒚 𝒋Ƹ + 𝑨𝒛 𝒌 components.
Example:
𝑨 = 𝟑𝒊Ƹ + 𝟐𝒋Ƹ = 𝟑, 𝟐
= 𝟔, −𝟏𝟎, 𝟏
𝑩 = 𝟔𝒊Ƹ − 𝟏𝟎𝒋Ƹ + 𝒌
𝑪 = 𝟏𝟎𝒋Ƹ + 𝒌 = 𝟎, −𝟏𝟎, 𝟏
For a vector 𝐴Ԧ in a 2-dimensional space, the magnitude can be calculated as:
𝑨 = 𝑨𝟐𝒙 + 𝑨𝟐𝒚
For a vector 𝐴Ԧ in a 3-dimensional space, the magnitude can be calculated as:
𝑨 = 𝑨𝟐𝒙 + 𝑨𝟐𝒚 + 𝑨𝟐𝒛 // Just think of it
this way: to get the
magnitude of a
Try these! vector, just get the
Find the magnitude of the following vectors: square root of the
1. <3,5,2> sum of the
2. 𝑖 − 𝑗 + 2𝑘 squares of the
3. <0,0,9> components ☺
For a vector 𝐴Ԧ in a 2-dimensional space, the magnitude can be calculated as:
−𝟏
𝑨𝒚
𝜽 = 𝐭𝐚𝐧
𝑨𝒙
Try these! TAKE NOTE!
Find the angle of inclination of the following vectors: The angle theta is actually the
1. <1.13, -3.22> angle of inclination of a vector with
respect to the positive x-axis. For a
2. <15.02, 29.16>
vector in 3D space, the angle theta
3. 4i + 7j does not completely describe its
4. <5,0> direction.
To find a unit vector, 𝒖, in the same direction of a vector, 𝑨 , we use the formula:
𝑨 // Simply get the ratio of the
𝒖= vector itself and its magnitude.
𝑨
Example: Find the unit vector in the same direction of the vector 𝐹Ԧ = < 3, −4 >
Example: Find the unit vector in the same direction of the vector 𝑉 = 𝑖Ƹ + 2𝑗Ƹ − 𝑘
