Dot Product and Force Vectors | Mechanics Statics | (Learn to solve any question)

Summary

This video script explores the application of dot products in calculating angles between vectors and projections of forces in 3D space. It explains how to determine the angle using the dot product formula and how to find the parallel and perpendicular components of a vector to a line. The script guides through examples involving triangular plates and force projections along cables, emphasizing the importance of expressing forces in Cartesian form and calculating unit vectors. It also covers converting non-Cartesian forces into Cartesian form for projection calculations, providing a comprehensive understanding of vector operations in physics.

Takeaways

• 📚 The dot product is a useful tool for calculating the angle between two forces or sides in 3D space.
• 🔍 To find the angle, use the dot product formula involving the magnitudes of the vectors and the cosine of the angle.
• 📐 Understanding the components of a vector, both parallel and perpendicular to a line, is essential for vector analysis.
• 📈 The parallel component of a vector can be found using the dot product with the unit vector of the axis.
• 📉 The perpendicular component can be determined using the Pythagorean theorem, with the given magnitude of the force.
• 📝 Expressing forces in Cartesian form and writing position vectors along with unit vectors are fundamental skills for these calculations.
• 🔄 Review previous material if needed, as the script assumes a certain level of understanding from a previous video.
• 📌 When calculating angles between sides of a triangular plate, use position vectors from two points and the dot product.
• 📐 For projection problems, identifying unit vectors for the lines of interest is a crucial step.
• 📊 The magnitude of the projected components of a force can be found by taking the dot product with the unit vector of the line.
• 📘 If a force is not given in Cartesian form, it must first be converted using a position vector and then finding the unit vector for the line.
• 🔑 The script provides a structured approach to solving problems involving dot products, angles, and projections in vector analysis.

Q & A

What is the dot product and how is it used in calculating the angle between two vectors?

The dot product is an algebraic operation that takes two equal-length vectors and returns a single number. It is used to calculate the angle between two vectors by using the formula: cos(θ) = (A · B) / (||A|| ||B||), where A and B are the vectors, θ is the angle, and ||A|| and ||B|| are the magnitudes of the vectors.

How do you find the components of a vector that are parallel and perpendicular to a line?

To find the parallel component, you take the dot product of the vector with the unit vector of the axis or line. The perpendicular component can be found using the Pythagorean theorem, considering the magnitude of the original vector, the parallel component, and the perpendicular component as parts of a right triangle.

What is the projection of a force and how is it found?

The projection of a force is the component of that force that is parallel to a given line or axis. It is found by taking the dot product of the force vector in Cartesian form with the unit vector of the axis or line.

Why is it necessary to express forces in Cartesian form when calculating projections?

Expressing forces in Cartesian form allows for the straightforward application of the dot product formula, which requires the vectors to be broken down into their x, y, and z components for calculation.

What is a unit vector and how is it used in finding projections?

A unit vector is a vector with a magnitude of one, used to specify direction. It is used in finding projections by being the reference direction for the dot product calculation, which helps in determining the component of a force along a particular axis or line.

How do you calculate the magnitude of a position vector?

The magnitude of a position vector is calculated using the formula ||v|| = √(x^2 + y^2 + z^2), where x, y, and z are the components of the vector.

What is the significance of the negative sign in the projection calculation?

The negative sign in the projection calculation indicates that the component points in the opposite direction to the unit vector, which is important for understanding the actual direction of the projection.

How does one find the perpendicular component of a vector using the Pythagorean theorem?

The perpendicular component can be found using the Pythagorean theorem by first determining the parallel component and then using the formula: perpendicular component = √(magnitude of force^2 - (parallel component)^2).

What is the process of expressing a force in Cartesian form when it is not initially given?

To express a force in Cartesian form when it is not initially given, one must first identify the relevant position vectors, calculate their magnitudes, and then find the unit vector. Multiplying the magnitude of the force by this unit vector gives the force in Cartesian form.

Can you provide an example of how to calculate the angle between two sides of a triangular plate using the dot product?

Sure. First, identify the position vectors for the two sides of the triangle. Calculate the magnitudes of these vectors. Then, compute the dot product of these vectors. Finally, use the formula cos(θ) = (dot product) / (magnitude of vector 1 * magnitude of vector 2) to find the angle θ.

What is the role of the magnitude in the dot product calculation for projections?

The magnitude of the force and the unit vector are essential in the dot product calculation for projections as they provide the scale for determining the actual length of the projected component along the axis or line.

Outlines

00:00

📚 Vector Dot Product and Angle Calculation

This paragraph introduces the concept of using the dot product to calculate the angle between two forces or sides in 3D space. It explains the formula for the dot product in terms of angle calculation and highlights the importance of expressing vectors in Cartesian form, as well as understanding position and unit vectors. The paragraph also touches on finding the parallel and perpendicular components of a vector with respect to a line, using the dot product and the Pythagorean theorem. It suggests revisiting a previous video for a step-by-step guide on these concepts and encourages moving on to examples to solidify understanding.

05:02

📐 Applications of Dot Product in Vector Projections

The second paragraph delves into practical applications of the dot product, specifically in calculating the magnitudes of projected components of a force along different vectors, such as cables AB and AC in a given problem. It outlines the steps to find unit vectors for the cables, the process of expressing a force in Cartesian form when not initially given, and the method to calculate the projection of a force onto a specific cable. The paragraph also addresses how to find the perpendicular component of a force using the Pythagorean theorem, providing a comprehensive guide to solving problems involving dot products and vector projections.

Keywords

💡Dot Product

The dot product, also known as the scalar product, is a binary operation that takes two vectors and returns a single number. In the context of the video, the dot product is used to calculate the angle between two vectors, which is essential in determining the relationship between two forces or sides in 3D space. The script mentions using the dot product to find the angle between the sides of a triangular plate, demonstrating its application in geometry and physics.

💡Angle

An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex. In the video, the angle is a key concept for understanding the spatial relationship between vectors. The script discusses using the dot product to find the angle between two sides of an object in 3D space, which is crucial for analyzing the forces acting on an object.

💡Vectors

A vector is a quantity that has both magnitude and direction. In the script, vectors are used to represent forces or sides of geometric shapes in 3D space. The video explains how to calculate the angle between two vectors using the dot product, and how to find the components of a vector that are parallel and perpendicular to a line, which are essential for understanding forces and their effects.

💡Projection

In the context of the video, projection refers to the component of a vector that lies along a particular line or axis. The script explains how to find the projection of a force onto a cable using the dot product between the force vector and the unit vector of the cable. This concept is important for understanding how much of a force is acting in a specific direction.

💡Unit Vector

A unit vector is a vector with a magnitude of one, used to specify a direction in space. In the video, unit vectors are derived from position vectors to represent the direction of cables AB and AC. The script demonstrates how to calculate unit vectors and use them in the dot product to find the projection of a force.

💡Cartesian Form

Cartesian form, or Cartesian coordinates, is a method of specifying points in a plane or space using a set of numbers. In the script, forces are expressed in Cartesian form to facilitate calculations, such as finding projections. The video explains how to convert a force given in a different form into Cartesian form for further analysis.

💡Magnitude

The magnitude of a vector is its length, which represents the size of the vector without considering its direction. In the video, the magnitude of position vectors and forces is calculated to use in the dot product formula and to determine the unit vectors. The script uses the magnitude to find the projections of forces onto cables.

💡Position Vector

A position vector is a vector that represents the position of a point in space relative to an origin. In the script, position vectors from point A to points B and C are used to define the sides of a triangle and to calculate the angle between these sides using the dot product.

💡Parallel Component

The parallel component of a vector is the part of the vector that lies along a given line or axis. The video script explains how to find the parallel component of a force to a cable by taking the dot product of the force with the unit vector of the cable, which is essential for understanding the force's effect along the cable.

💡Perpendicular Component

The perpendicular component of a vector is the part of the vector that is at a right angle to a given line or axis. In the video, the perpendicular component is found using the Pythagorean theorem after determining the parallel component. The script uses this concept to analyze the force's effect that is perpendicular to the cable.

💡Pythagorean Theorem

The Pythagorean theorem is a principle in geometry that states the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. In the script, the theorem is used to calculate the perpendicular component of a force after finding the parallel component, illustrating its application in vector analysis.

Highlights

Calculating the angle between two forces or sides in 3D space can be achieved using the dot product.

The dot product formula can be rearranged to find the angle between two vectors.

Finding the components of a vector parallel and perpendicular to a line involves the concept of projection.

The parallel component of a vector can be found using the dot product with the unit vector of the axis.

The perpendicular component of a vector is calculated using the Pythagorean theorem.

Expressing forces in cartesian form and writing position vectors are essential for solving problems involving vectors.

A refresher on expressing forces and position vectors can be found in the previous video's description.

Examples are used to illustrate how to find angles between sides of a triangular plate using the dot product.

Position vectors from points A to B and A to C are required to calculate the angle between the sides of a triangle.

The magnitude of position vectors is necessary before taking the dot product.

The dot product results in a scalar value which is then used to find the angle between vectors.

Unit vectors are essential for finding the projections of forces along specific directions.

The projection of a force onto a cable is found by taking the dot product with the cable's unit vector.

Forces must be expressed in cartesian form when calculating projections, if not already given.

The process of finding unit vectors involves calculating the magnitude of position vectors and normalizing them.

The perpendicular component of a force can be found using the Pythagorean theorem after determining the parallel component.

Understanding dot products and vector components is crucial for solving physics problems involving forces.

Transcripts

Sometimes, we will get situations where we  need to calculate the angle between 2 forces,  

or 2 sides on an object in 3D space. In that  case, using the dot product can be really helpful.  

So if we have 2 vectors with an angle,  we can write the dot product like this.  

If you’re trying to find the angle, we can write  the equation like this to calculate the angle  

between the 2 sides. In another situation, we  might need to find the components of a vector  

that is parallel and perpendicular to a line.  The parallel component is sometimes shown like  

this, or like this, and its referred to as the  projection of a force. To find it, all we do is  

take the dot product between the force expressed  in cartesian form and the unit vector of the axis,  

or side where the projection is supposed to be.  If we need to find the perpendicular component,  

then we can use this equation to do it,  which is just the Pythagorean theorem.  

Here, A is the magnitude of the force, this is the  parallel component and this is the perpendicular  

component. This will become much clearer with  examples, but to do so, you should be able to  

express forces in cartesian form, and be able to  write position vectors along with unit vectors.  

In the previous video, I go through this  step by step, so if you need a refresh,  

or you forgot, please check the description.  Now let’s move on to some examples, and see  

how we can find angles, projections and more. Let’s take a look at this question where all we  

need to do is figure out the angle between the  sides of the triangular plate. We are going to  

do it using the dot product. So let’s write  down the equation to figure out an angle. 

So first, we need 2 position vectors, for the 2  sides of the triangle. One from A to B and one  

from A to C. Let’s write down the locations of  points A, B and C (pause after each letter). Now  

we can write our first position vector. Let’s  simplify. Next, a position vector from A to C. 

Now, we need to find the magnitude of  these position vectors. Pause 2 seconds. 

Next, we will take the dot product between  the 2 position vectors. So all we’re doing  

is multiplying I components by I components, j  components by j components and k components by k  

components. The result is a scalar value. Now all that is left is to plug everything  

into our equation. Solving gives us our answer. Let’s take a look at this question involving  

projections. We need to figure out the magnitudes  of the projected components of the force along  

cables AB and AC. When we have questions asking  for projections, we need to find unit vectors.  

So in this problem, we need a unit vector  for cable AB and cable AC. So let’s start  

by writing down the locations of each point.  So we have points A, B and C. Next, we need  

position vectors from A to B and A to C. Now we  can find the magnitude. Okay, so finally, we can  

divide each component in our position vector  by the magnitude, and we get our unit vectors. 

We are now ready to do the dot product. First,  the projection of force F onto cable AB. For that,  

all we do is take the dot product between force  F and the unit vector AB. Remember, force F  

has to be expressed in cartesian form, and since  the question already gives it in cartesian form,  

we can just take the dot product. Since  we are asked for the magnitude, remember,  

it’ll be positive. The negative sign means that  our components points in the opposite direction  

to the unit vector. Next, the projection of  force F onto cable AC. Those are our answers. 

Let’s take a look at this problem where we need  to find the projected component of the 600 N force  

along rod AC. Unlike the previous question,  our force isn’t given in cartesian form,  

so we need to express it as such. After  that, we need a unit vector for rod AC.  

So do to all this, we first need to write down  the locations of all the points of interest.  

First, point A. Next, we need to figure out  point B. The question says point B is at  

the midpoint of rod AC. So the x component  would be 1.5m in the negative x direction,  

the y component would be 2m and the zed component  would also be 2m. We are just dividing the  

distances by 2. Next, we have points C and D. To express the 600 N force in cartesian form,  

we need a position vector from B to D. Now we need the magnitude of this vector. 

Next step is to divide each component  in our position vector by the magnitude,  

which gives us our unit vector. Lastly, we just multiply the magnitude  

of the force by the unit vector, and that  is our force expressed in cartesian form. 

Now we need a unit vector for rod AC. The process  

is pretty much the same. First,  a position vector from A to C. 

Now the magnitude. And lastly, the unit vector.  

Let’s write it in decimal form. Now all that’s left is to take the  

dot product between the force and the unit  vector. Solving this gives us the component  

of the force that is along rod AC. Now we  need the perpendicular component. For that,  

we use this equation. So F is the magnitude of our  Force, so it’s 600 N. The parallel component is  

what we just found, so now, we can solve for the  perpendicular component. Those are our answers. 

That should cover the types of problems you will  face when it comes to dot products and forces.  

Thanks for watching and best  of luck with your studies!