AQA A’Level Vectors - Part 5, Application of dot product

Summary

This 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

00:00

📐 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.

Keywords

💡Dot product

The dot product, also called scalar product, is a mathematical operation that multiplies two vectors and results in a scalar (a single number). In the video, it is used to find the angle between two vectors. The dot product is essential in applications like computer graphics and physics for calculating angles between objects.

💡Vectors

A vector is a quantity that has both magnitude and direction, commonly represented as an arrow. In this video, two vectors 'A' and 'B' are used to demonstrate the calculation of the angle between them using the dot product formula. Vectors are fundamental in various fields such as physics, engineering, and computer science.

💡Angle between vectors

The angle between two vectors is a key concept in vector mathematics. It is calculated by using the dot product and the magnitudes of the vectors. In this video, the angle between two vectors is found to be 45 degrees, which helps in understanding spatial relationships, especially in graphical applications.

💡Pythagoras' theorem

Pythagoras' theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The video uses this theorem to calculate the length of the vectors, which is an essential step before finding the angle between them.

💡Length of a vector

The length of a vector (also called its magnitude) is the distance from the vector's origin to its endpoint. It is calculated using Pythagoras’ theorem. In the video, the lengths of vectors A and B are computed to help calculate the angle between them using the dot product.

💡Graphical applications

Graphical applications refer to computer programs that deal with image creation, manipulation, and display. The video mentions that calculating angles between vectors using the dot product has significant applications in graphical programs and computer games, where object positioning and movement are important.

💡Scalar

A scalar is a single number used to describe quantities like mass, temperature, or speed, unlike a vector that has both magnitude and direction. The result of the dot product of two vectors is a scalar, as demonstrated in the video.

💡Scientific calculator

A scientific calculator is an advanced calculator capable of performing functions like trigonometry, logarithms, and complex equations. In the video, it is used to look up the cosine value and find the angle between the vectors, a crucial step in solving the example provided.

💡Cosine (COS)

Cosine (COS) is a trigonometric function that relates the adjacent side to the hypotenuse in a right-angled triangle. In the video, cosine is calculated from the dot product and the magnitudes of the vectors. The angle between the vectors is derived from the cosine value.

💡Protractor

A protractor is a geometric tool used to measure angles. In the video, after calculating the angle between vectors A and B, a protractor is used to verify that the calculated angle of 45 degrees is correct, demonstrating the accuracy of the mathematical process.

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