how to calculate displacement vectors?

Indranil

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In the diagram, If r1 and r2 position vectors in the diagram how to calculate the displacement vector d in another way?
d = r2 - r1 as shown in the diagram But how to calculate d using triangle law here and what will be the diagram? The first diagram is for d = r2 - r1 and I have put the second diagram for the triangle law by changing the arrow into red color. Please let me if my changed diagram is correct or wrong.
 

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You could find the angle subtended by the two vectors and apply the Law of Cosines to find the magnitude of the difference. Is that what you mean?
 
You could find the angle subtended by the two vectors and apply the Law of Cosines to find the magnitude of the difference. Is that what you mean?
Yes, exactly that's what I mean. Could you explain, please?
 
Yes, exactly that's what I mean. Could you explain, please?

Suppose \(\displaystyle \vec{r_1}\) has an angle of inclination \(\displaystyle \alpha\) and \(\displaystyle \vec{r_2}\) has an angle of inclination \(\displaystyle \beta\)...then the angle \(\displaystyle \theta\) subtended by the two vectors is given by:

\(\displaystyle \theta=\alpha-\beta\)

And so, applying the Law of Cosines, we may state:

\(\displaystyle \left|\vec{D}\right|=\sqrt{\left|\vec{r_1}\right|^2+\left|\vec{r_2}\right|^2-2\left|\vec{r_1}\right|\left|\vec{r_2}\right|\cos(\theta)}\)
 
Suppose \(\displaystyle \vec{r_1}\) has an angle of inclination \(\displaystyle \alpha\) and \(\displaystyle \vec{r_2}\) has an angle of inclination \(\displaystyle \beta\)...then the angle \(\displaystyle \theta\) subtended by the two vectors is given by:

\(\displaystyle \theta=\alpha-\beta\)

And so, applying the Law of Cosines, we may state:

\(\displaystyle \left|\vec{D}\right|=\sqrt{\left|\vec{r_1}\right|^2+\left|\vec{r_2}\right|^2-2\left|\vec{r_1}\right|\left|\vec{r_2}\right|\cos(\theta)}\)
Could you please put a diagram against your solution so that I can understand the concept easily?
 
You can refer to the diagram you posted yourself. What about what I posted wasn't clear?
 
… I have put the second diagram … by changing the arrow into red color.
Why did you reverse the direction of vector r1 with a second diagram? Are you trying to ask two different questions?
 
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Why did you reverse the direction of vector a, and post the second diagram? Are you trying to ask two different questions?
Yes, my question is if I want to calculate the displacement vector d by using the cosine rule, then what will be the diagram?
I have just tried myself by drawing the red arrow to make a triangle where I can calculate the displacement vector d (resultant vector). Please let me know If I have drawn the diagram correctly.
 
Is this your answer to my second question?


I don't understand. Please rephrase your answer. What does the red arrow represent, to you?

We have a language barrier.
If d = r2 - r1, where r2 = final position, r1 = initial position and d = displacement vector or resultant vector, then what will be the diagram if we use the cosine rule of vector? this is my question.
 
Using the "cosine law" to find the length of the resultant vector, the direction of the vectors is irrelevant. Only the angles and the lengths of the sides of the triangle are relevant.
 
… d = displacement vector

… then what will be the diagram if we use the cosine rule … this is my question.
I still do not understand why you posted four diagrams with coordinate axes.

Two of the diagrams show a line segment connecting points A and B. They labeled this line: vector d. But, they don't show its direction! Which end of vector d is the head? (Very sloppy.)

Additionally, they seem to be talking about position vectors. If they want to represent vector d as a position vector, then they have not correctly positioned it.

Regarding your question above, diagrams do not change if somebody chooses to use the Law of Cosines. Also, the Law of Cosines doesn't care whether you have a diagram, whether you're working with vectors or simply a triangle, or what you're thinking or doing otherwise. The Law of Cosines is just a way to find the length of the third side of a triangle.

This situation (contrasting diagrams with theorems) is the same concerning other theorems, like the Pythagorean Theorem, for example. A theorem doesn't care whether you have a drawing, what the drawing represents or why you drew it. AND, if you choose Pythagorean Theorem to find hypotenuse length (versus some other method), the diagram does not change!

I agree with Halls. We use the Law of Cosines to find the distance between points A and B. That length is not a vector. It is the magnitude of a vector.

I hope you find time to take a beginning class on vectors. You need both comprehension and experience, from a structured approach to learning. :cool:
 
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