AQA A’Level Vectors - Part 5, Application of dot product
Summary
TLDRThis video explores the practical application of the dot product in finding the angle between two vectors. It walks through a five-step process to calculate the angle, demonstrating its usefulness in fields like computer science, especially in graphics and gaming. Using an example, the video shows how to compute the dot product, determine vector lengths, and use trigonometric formulas to find the angle, arriving at 45 degrees. Viewers are encouraged to follow the steps slowly, pause as needed, and use tools like calculators or protractors for verification.
Takeaways
- 📚 This video focuses on the application of the dot product in vector mathematics.
- 🔍 The dot product is essential for finding the angle between two vectors.
- 💻 It's particularly useful in computer graphics and gaming applications.
- 📐 The formula to calculate the angle involves the dot product and the lengths of the vectors.
- 📝 A five-step process is outlined for calculating the angle between two vectors.
- 🧮 Step one involves calculating the dot product of two vectors.
- 📏 Step two requires calculating the lengths of each vector using Pythagoras' theorem.
- 🔢 Step three combines the dot product and vector lengths to form a part of the formula.
- 📉 Step four involves calculating the cosine of the angle using the formula components.
- 🔍 Step five is to find the angle by looking up the cosine value in tables or using a calculator.
- 📏 The example provided demonstrates calculating the angle to be approximately 45 degrees.
Q & A
What is the main topic of the final video in the series?
-The main topic of the final video is the application of the dot product to find the angle between two vectors.
Why is calculating the angle between two vectors important?
-Calculating the angle between two vectors is important because it has many applications in computer science, especially in graphical applications and computer games.
What mathematical tool is used to find the angle between two vectors?
-The dot product is used to find the angle between two vectors.
What is the five-step process mentioned in the video to calculate the angle between two vectors?
-The five-step process involves: 1) Calculating the dot product of the two vectors, 2) Finding the lengths of each vector using Pythagoras' theorem, 3) Multiplying the two lengths together, 4) Dividing the dot product by the product of the vector lengths, and 5) Using a calculator or table to find the angle.
How is the dot product of two vectors calculated in this example?
-The dot product is calculated by multiplying the corresponding components of the two vectors and then adding the results. For example, 4 * 8 = 32 and 9 * 3 = 27, and then 32 + 27 = 59.
How are the lengths of vectors A and B calculated?
-The lengths of vectors A and B are calculated using Pythagoras' theorem by squaring the components of each vector, summing the squares, and then taking the square root of the sum.
What is done in step three of the process?
-In step three, the lengths of vectors A and B, calculated in step two, are multiplied together.
How is the cosine of the angle between the two vectors calculated?
-The cosine of the angle is calculated by dividing the dot product of the two vectors (59) by the product of their lengths (84.12).
What value is obtained for cos(θ) in this example?
-The value of cos(θ) obtained is approximately 0.701.
How is the angle between the vectors found after calculating cos(θ)?
-The angle is found by looking up the cosine value (0.701) in a table or using a scientific calculator, which gives an angle of approximately 45 degrees.
Outlines
📐 Introduction to Dot Product Application
This paragraph introduces the importance of the dot product in vector mathematics, particularly its application in finding the angle between two vectors. It emphasizes the utility of this concept in computer science, especially in graphical applications and computer games. The paragraph outlines a five-step process to calculate the angle between two vectors using the dot product formula. The steps include calculating the dot product of the vectors, determining the lengths of the vectors using Pythagoras' theorem, and then using these values to find the cosine of the angle between them. The process concludes with looking up the angle in tables or using a scientific calculator.
Mindmap
Keywords
💡Dot product
💡Vectors
💡Angle between vectors
💡Pythagoras' theorem
💡Length of a vector
💡Graphical applications
💡Scalar
💡Scientific calculator
💡Cosine (COS)
💡Protractor
Highlights
Introduction to the dot product and its importance in finding the angle between vectors.
The dot product has many applications, especially in computer science and graphical programs, including video games.
The formula for calculating the angle between two vectors is introduced.
A five-step process is provided for calculating the angle between vectors using the dot product.
Step 1 involves calculating the dot product by multiplying corresponding components of vectors and summing the results.
Example provided for calculating the dot product of vectors A and B, with specific values resulting in 59.
Step 2 involves calculating the length of each vector using Pythagoras' theorem.
An example demonstrates how to calculate vector lengths using squared components and the square root.
Step 3 involves multiplying the lengths of the two vectors to complete part of the angle formula.
The combined vector length calculation results in a value of 84.12.
Step 4 shows the division of the dot product by the multiplied vector lengths to find cos(θ).
The resulting value for cos(θ) is approximately 0.2701.
Step 5 involves using a scientific calculator or tables to find the corresponding angle for cos(θ).
The example concludes with the angle between vectors A and B being approximately 45 degrees.
The accuracy of the calculation is confirmed by checking the angle using a protractor.
Transcripts
in this final video on vectors we're
going to look at one of the most
important applications of dot product if
you haven't done so already go back and
watch the other videos on vectors in
this series first okay so in the last
video we looked at how to produce a dot
or scalar product but what's the point
of calculating the dot product of two
value of two vectors well it can be used
to find the angle between any two
vectors that you supply this has many
applications in computer science but can
be especially useful in graphical
application programs and computer games
as is shown here by this screenshot
okay so let's actually try calculating
the angle between the two vectors a and
B in order to do this we have to use
this formula where a and B of two
vectors that have been supplied and
where the lengths of the vectors a and B
can be written in this mathematical
notation there's a five-step process you
have to follow you have to calculate the
dot products the two vectors calculate
the lengths of each vector by using
Pythagoras theorem times the length of
the two vectors together calculate COS
and then look up the value in tables in
order to get the resultant angle let's
work through an example step by step we
can leave the five-step process at the
top here take your time and pause the
video as much as you need to work
through this slowly so step one is
calculating the dot product of vectors a
and B now remember part four the video
in this series shows you how to do this
but in essence we take each value from
each vector multiply them together and
then add the results so we start by
taking the four from this vector and the
eight from this vector multiplying them
together to get 32 we then take the nine
from this vector and the three from this
vector and we hadn´t multiply them
together to get 27 we add the results
together to get 59 so now we have the
dot product of vectors a and B that's
the top part of this formula done so the
dot product on we now have to calculate
the length of vector a and vector B and
for this we can simply use Pythagoras
theorem so we can take the vector four
were nine square them at the results
perform a square root to find out the
length of the first vector we do exactly
the same to vector B to get the length
of the second vector now we have the
length of vector a and vector B we move
on to step three which is two times the
two numbers together to get the number
84 point one two and that's now this
part
of the formula step for them is to work
out this section here and of course we
have the values 59 worked out from step
one and the value is eighty four point
one two worked out from step two and
three and the result of this number
divided by this number is naught point
two seven zero one then we can move on
to step 5 step five then we can simply
look up the value of this in tables or
if you have a scientific calculator you
can look it up in there and as we can
see naught point seven zero one or close
enough is approximately 45 degrees so
this angle here should be 45 degrees we
can physically check that by using a
protractor and you can see here that
spot on the angle of vectors a and B is
45 degrees
you
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