SHS General Physics 1 | Lesson 3: VECTOR ADDITION
Summary
TLDRIn this lesson, students will learn about vectors, including differentiating vectors and scalar quantities, performing vector addition, and rewriting vectors in component form. The lesson begins with a review of trigonometric functions and the law of cosines. It covers vector representation, addition of parallel and non-parallel vectors, and methods like the tip-to-tail and parallelogram methods for finding resultant vectors. The lesson also includes numerical analysis for calculating resultant forces and introduces unit vectors and their calculations. Practical examples and problems are provided for better understanding. Students are assigned activities to reinforce the concepts learned.
Takeaways
- 📚 The lesson covers vectors, including differentiation between vectors and scalars, vector addition, and writing vectors in component form.
- 📐 Quick review of SOHCAHTOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.
- 🔢 Example problem solving: For a 30-degree triangle with given sides, calculating sine, cosine, and tangent.
- 🔺 Law of cosines: Useful for finding a third side of a triangle when two sides and the angle between them are known, or finding angles when all three sides are known.
- 🛠️ Scalars are quantities described by magnitude only, such as speed, volume, mass, and time.
- ➡️ Vectors have both magnitude and direction, important in motion studies; examples include force, velocity, acceleration, and momentum.
- 📏 Representation of vectors: Typically a letter with an arrow above or in boldface, and the magnitude is shown without an arrow or with vertical bars.
- ➕ Vector addition: Adding corresponding components of vectors to find the resultant vector.
- 🔄 Adding parallel and non-parallel vectors: Use direction consideration for parallel vectors and tip-to-tail or parallelogram methods for non-parallel vectors.
- 📊 Numerical analysis: Law of cosines and Pythagorean theorem can be used for calculating resultant forces in non-graphical methods.
Q & A
What are the objectives of today's lesson on vectors?
-The objectives are to differentiate vectors and scalar quantities, perform addition of vectors, and rewrite a vector in component form.
What is SOHCAHTOA and how is it used?
-SOHCAHTOA is a mnemonic to remember the definitions of sine, cosine, and tangent: Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.
How do you solve for sine, cosine, and tangent in a 30-degree triangle with given side lengths?
-For sine, divide the opposite side (1) by the hypotenuse (2) to get 0.5. For cosine, divide the adjacent side (√3) by the hypotenuse (2) to get 0.866. For tangent, divide the opposite side (1) by the adjacent side (√3) to get 0.577.
What is the law of cosines and when is it useful?
-The law of cosines is useful for finding the third side of a triangle when two sides and the angle between them are known, or for finding the angles when all three sides are known. The formula is c² = a² + b² - 2ab * cos(C).
What is the difference between a scalar and a vector quantity?
-A scalar quantity is described by a magnitude only, while a vector quantity has both a magnitude and a direction.
How are vectors typically represented?
-Vectors are usually represented by a letter with an arrow above it or in boldface. The magnitude of a vector can be represented by a lightface letter without an arrow or the vector symbol with vertical bars on both sides.
How do you add two vectors in component form?
-To add two vectors in component form, you add their corresponding components. For example, vector A (2, 4) + vector B (-1, 6) results in vector C (1, 10).
How do you add parallel vectors?
-For parallel vectors in the same direction, you add their magnitudes. For vectors in opposite directions, you subtract their magnitudes.
What methods can be used to add non-parallel vectors?
-The tip-to-tail method and the parallelogram method can be used to add non-parallel vectors.
How can you calculate the resultant force using the law of cosines?
-Using the law of cosines, you can calculate the resultant force by applying the formula: R² = a² + b² - 2ab * cos(C). Then, solve for R.
What is a unit vector and how is it calculated?
-A unit vector has a magnitude of one and is used to indicate direction. It is calculated by dividing each component of the vector by the vector's magnitude.
How can you find the direction of a vector using trigonometry?
-The direction of a vector can be found using the arctangent function: θ = arctan(b/a), where b and a are the components of the vector.
What is the process for finding the net force when multiple forces act at different angles?
-To find the net force, break each force into its x and y components, sum the components separately, then use the Pythagorean theorem to find the resultant magnitude and arctangent to find the direction.
What activities were assigned at the end of the lesson?
-The activities assigned are: Activity 1 on page 13, Activity 2 on page 14, Activity 3 on page 15, and Activity 4 on page 16.
Outlines
🎓 Introduction to Vectors and Scalars
This lesson covers vectors and scalar quantities. By the end, students will be able to differentiate between vectors and scalars, perform vector addition, and rewrite vectors in component form. A review of SOHCAHTOA helps refresh knowledge on sine, cosine, and tangent functions. The lesson also recalls the law of cosines, useful for finding triangle sides or angles. Scalars, described by magnitude alone, include quantities like speed and mass. Vectors, which have both magnitude and direction, include force and velocity.
➕ Vector Representation and Addition
This section explains how vectors are represented, with arrows or bold letters for vectors and lightface letters or bars for magnitudes. It discusses vector components, which are ordered pairs describing changes in x and y values. Vectors are equal if they have the same magnitude and direction. Vector addition is performed by adding corresponding components. Examples include adding vectors and handling parallel vectors by considering their directions, resulting in net forces based on their magnitudes and directions.
📐 Methods for Adding Non-Parallel Vectors
This part covers methods for adding non-parallel vectors using the parallelogram method and tip-to-tail method. An example is given with forces of 75N and 50N at a 60-degree angle. The parallelogram method involves redrawing diagrams to scale and finding the resultant force as the diagonal. The tip-to-tail method involves drawing arrows representing forces and finding the resultant by connecting the starting point of the first force to the endpoint of the second. Numerical analysis is introduced to calculate resultants using the law of cosines.
📏 Numerical Analysis of Vectors
This section illustrates solving vector problems numerically. For example, calculating the resultant force using the law of cosines when given two sides and an angle. Another example involves a weight of 8N pulled sideways by a 5N force, using the Pythagorean theorem to find the resultant. Unit vectors, which have a magnitude of one, are explained. Calculations for unit vectors in two and three dimensions are demonstrated, and vector addition/subtraction using components is shown. Magnitude and direction of vectors can be calculated using specific formulas.
⚖️ Advanced Vector Problems and Activities
This final section presents complex vector problems, such as calculating the net force from multiple forces at different angles. It involves breaking forces into x and y components and using trigonometric functions to determine these components. The resultant is found by summing the components and applying the Pythagorean theorem. The direction is found using the arctan function. The section concludes with assigned activities from the textbook, reinforcing the lesson's concepts through practice problems.
Mindmap
Keywords
💡Vectors
💡Scalars
💡Addition of Vectors
💡Component Form
💡Resultant
💡Parallelogram Method
💡Tip-to-Tail Method
💡Law of Cosines
💡Unit Vector
💡Pythagorean Theorem
💡Magnitude
Highlights
Differentiate between vectors and scalar quantities.
Perform addition of vectors.
Rewrite a vector in component form.
Recall the basic trigonometric functions: sine, cosine, and tangent.
Example problem involving a 30-degree triangle with specific side lengths.
Explanation of the law of cosines for finding sides or angles of a triangle.
Definition and examples of scalar quantities like speed, volume, mass, and time.
Definition and examples of vector quantities like force, velocity, acceleration, and momentum.
Representation of vectors using letters with arrows or boldface letters.
Component form of a vector as an ordered pair describing changes in x and y values.
Methods for adding vectors: summing corresponding components and graphical methods like the parallelogram method.
Adding parallel vectors by considering direction: sum for the same direction, subtract for opposing directions.
Non-parallel vector addition using the tip-to-tail method and the parallelogram method.
Example problem using the parallelogram method to find the resultant of two non-parallel forces.
Introduction of unit vectors as vectors with a magnitude of one unit, often used to denote direction.
Algebraic addition and subtraction of vectors using their components.
Calculating the magnitude and direction of a vector using the Pythagorean theorem and trigonometric functions.
Example problem involving three forces acting on a point, requiring summation of x and y components.
Transcripts
[Music]
hello dear students
our lesson for today is all about
vectors
at the end of this lesson you will be
able to
1. differentiate vectors and scalar
quantities
two perform addition of vectors and
three
rewrite a vector in component form
before we start
let us have a short review about sokodoa
we all know that it is just an easy way
to remember how sine
cosine and tangent works for sine it is
always opposite over hypotenuse
for cosine it is adjacent over
hypotenuse
and for tangent it is opposite over
adjacent
to refresh your knowledge let us answer
this simple problem
a 30 degree triangle has a hypotenuse of
length 2
an opposite side of length 1 and an
adjacent side of square root of 3
as shown on the figure now try solving
for the functions
for sine we get 0.5 by dividing opposite
magnitude of 1 with the hypotenuse
magnitude of 2.
for cosine we get 0.866 by dividing the
adjacent magnitude of square root of 3
with the hypotenuse magnitude of 2.
lastly for tangent we get 0.577
by dividing the opposite magnitude of 1
with the adjacent magnitude of square
root of 3.
for this lesson it is also important
that we recall the law of cosines
the law of cosines is useful for finding
either
the third side of a triangle when we
know two sides and the angle between
them
or the angles of a triangle when we know
all three sides
let us now continue with our lesson
vectors and scalars
a scalar is a quantity that is fully
described by a magnitude only
it is described by just a single number
some examples of scalar quantities
include speed
volume mass and time on the other hand
a vector is a quantity that has both a
magnitude and a direction
vector quantities are important in the
study of motion
some examples of vector quantities
include force velocity
acceleration and momentum the following
are the parameters considered as vectors
lift displacement weight drag force
momentum acceleration and velocity
for scalar we have time distance mass
volume area density work temperature
speed energy and power representation of
vectors
a vector is usually represented by
either a letter with an arrow above the
letter or a bold face letter
the magnitude of a vector is represented
by either a light face letter without an
arrow on top
or the vector symbol with vertical bars
on both sides
component of a vector the component form
of a vector is the ordered pair that
describes the changes in the x and y
values
we can say that two vectors are equal if
they have the same magnitude and
direction
or when they are parallel if they have
the same or opposite direction
we can combine vectors by adding them
the sum of two vectors is called the
resultant
in this picture the resultant is the
arrow between the vectors and b
in order to add two vectors we add the
corresponding components
example add the two following vectors
vector a with ordered pair two and four
and vector b with ordered pair negative
one and six
we add the corresponding components as
shown
vector of plus vector b equals two plus
negative one
and four plus six the answer is one and
ten
adding parallel vectors to add parallel
vectors
we just need to consider the direction
we add vectors in the same direction
and subtract those who oppose each other
for the first example
the forces 1 newton and 1.5 newton are
both directed to the right
so the f net will be one plus 1.5 equals
to 2.5 newtons to the right
for the second example the forces 1
newton and 1.5 newton are both directed
to the left
while 1.5 newton is directed to the
right
the f net will be 1.5 plus 1 minus 1.5
we get 1 newton to the left
non-parallel forces to add non-parallel
forces we use two methods
tip to tail method parallelogram method
let us answer this example while
illustrating how to use the two methods
consider two forces that are not acting
along the same line on an object
find the resultant force f1 is 75
newtons
while f2 is 50 newtons the angle between
them is 60 degrees
using the parallelogram method step 1
redraw the given diagram using a
suitable scale to represent the forces
with arrows
note 1 centimeter represents 10 newtons
so 75 n equals 7.5 centimeters and 50 n
equals 5 centimeters
step 2 complete a parallelogram to scale
step 3 the resultant force is
represented by the diagonal of the
parallelogram
find the magnitude and direction of the
resultant force to scale
measuring the length of the yellow line
we get f net equals 10.6 times
10 equals 106n at an angle of 25 to the
power of 0 from 75n
tip to tail method step 1 draw an arrow
to represent one of the two forces using
a suitable scale
note 1 centimeter represents 10 newtons
so 50 n equals 5 centimeters
step 2 where the first arrow ends draw
another arrow to represent the second
force 75n so that the tip of the first
arrow joins the tail of the second arrow
step 3 the resultant force is found by
joining the starting point of first
force to the endpoint of second force
find the magnitude and direction of the
resultant force to scale
measuring the resultant represented by
the yellow line
f net equals 10.6 times 10 equals 106n
at an angle of 35 to the power of 0 from
50n
[Music]
you may have observed that the recent
methods are graphical and you may be
worried that it will consume time
however we can calculate the resultant
numerically so you have nothing to worry
for the numerical analysis first observe
the figure on the bottom
the connected lines formed a triangle
which means
we can use any method that we know can
solve a triangle
look closely we are given two sides and
an angle between the adjacent side and
the hypotenuse
[Music]
through the given we can conclude that
we can use the law of cosines
the formulas for the law of cosines are
shown in the picture
now using the cosine law we can compute
for the resultant
r r squared is equals to the square of
adjacent side
plus the opposite side minus the product
of twice the two sides and cause of the
angle opposite the hypotenuse
r that we are looking for typing this on
your calculators we get 108.97
next example a weight of eight newtons
hangs from the end of a row
it is pulled sideways by a horizontal
force f of five
n and is held stationary what is t
we can draw the figure using the
parallelogram method and the tip-to-tail
method
like what is shown in the picture below
to solve this using a numerical analysis
let us focus on figure enclosed in a red
shape
looking closely observe that the shape
formed by connecting the lines is a
right triangle
because of this we can use the
pythagorean theorem to solve for the
resultant
just square the sides and add them then
get the square root of their sum
we get t equals 9.4339
unit vector a unit vector is a vector
that has a magnitude of one unit
a unit vector is also known as a
direction vector
the symbol for the unit vector is
usually a lower case letter with a hat
such as shown in the picture the unit
vector of a vector can be calculated
using the given formula
unit vector is equals to the ratio of 1
and the magnitude of the given vector
where the magnitude of the vector can be
calculated by getting the square root of
the sum of the squares of the ordered
pair
unit vectors can be used in two
dimensions here we show that the vector
a is made up of two
x unit vectors and 1.3 y unit vectors
likewise we can use unit vectors in
three or more
dimensions
[Music]
we can add or subtract vectors
algebraically using the vector
components
using the given vectors let us try a
minus c
a c equals 4 sub x i plus 5 sub z k
minus 7 sub y j minus 8 sub z k
reordering the components with the same
variables we arrive at 4 sub x minus 7
sub j
plus 13 sub z k
[Music]
we can compute for the magnitude of the
vector and find its direction using the
formula shown
for the direction we use theta is equals
to arc tan of b over a
[Music]
let us try computing for the magnitude
and direction of vector a
as you may have observed the magnitude
is calculated just like the pythagorean
theorem
substituting the values we get magnitude
of vector
as 6.4 units for its direction
simply get the arc tan of five over four
we get theta as 51.34
let's try another sample problem three
forces act on a point
eight n at zero degrees 9n at 90 degrees
and 10n at 217 degrees what is the net
force
first let us draw the diagram for this
problem so we can easily analyze
for 8 newtons it is at 0 degrees so we
draw it like this
then let us draw 9 newtons which is
directed to 90 degrees
third 10 newtons is at 217 degrees
finally let us draw the resultant
observe that the shaped formed is not a
triangle
not even a square so we cannot use the
pythagorean theorem and the law for
cosines
to solve this let us consider the
summation of the foss's x and y
components
for the summation of the foss's x
component we write f sub x
all magnitudes directed to the right are
positive while all that is directed to
the left will be negative
we write f x as equals to positive 8
plus 0
because the 9 newton force has no x
component
to get fx remember that the position of
of the force 10 newtons is at 217
degrees
now locating the x component we draw a
yellow dashed line and a red dashed line
for the y component
to be able to use sine we need to get
the angle between the hypotenuse and the
y component
subtracting 217 from 270 we get 53
now we can write fx equals 10 sine 53
the summation of fx is 0.013644 newtons
for the summation of fy we write all
magnitudes directed upward as positive
now to get fy we use cosine considering
that the angle is between the adjacent
side and the hypotenuse
for the summation of fy we get 2.9818
now to get the resultant we simply
compute for the square root of the
square of the summation of fx and fy
our resultant r is equals 2.9818 newtons
to determine whether our resultant's
direction is right let us compute for
theta arctan multiplied by fy
over fx we get 89.7378 degrees
[Music]
for your activities answer activity 1 on
page 13
activity 2 on page 14 activity 3 on page
15 and activity 4 on page 16.
that's all for today have a nice day
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