A few questions relating to the cross product

[Spiffy]

Ug. My first try at this and my whole post was deleted. Here goes again :)

Homework Statement

i) Two cities on the surface of the Earth are represented by position vectors that connect the location of each city to the centre of the earth. Assuming that the centre of the Earth is assigned the coordinates of the origin, and that the Earth is a perfect sphere, outline the steps that would lead to a calculation of the shortest distance between the two cities. Hint: How can you determine arc length?

ii) Prove A is perpendicular to B if |A+B| = |A-B|

iii) A||B if AxB=0

The Attempt at a Solution

i) well you have two vectors pointing outword tail to tail and an angle Θ in between them. What I don't really know is how to find the arc length as the hint wants me to. If someone could point me toward an equation i'll report with some progress

ii) It seems so obvious that I'm sure it's staring me in the face and I don't even know it. I have drawn on my paper A vector downward and B and -B vector going on at 90^o degrees either way. So now I have one triangle made up of two others and the two triangles are obviously equal to each other, they share adj and opp sidelengths and therefore the hypotenuse must as well be equal in both. I don't kknow how to take this from my sketch to something a bit more formal and algebraic though..

iii) I know that its true because if they were parallel sinΘ would = 0. I have:
AxB = |A||B|sinΘ
If A and B are parallel Θ=0
RS: |A||B|sin0
= 0
therefore AxB=0 when A and B are paralell. Would you accept this?

I appreciate any light that can be shed on some of these questions. I know this website is going to be a valuable resource for myself now and in the future. Even just typing questions up here is helping. (I originally had 4 questions but solved one just by putting it into a different form of media) thanks a lot!

 

[tiny-tim]

Welcome to PF!

Ug. My first try at this and my whole post was deleted. Here goes again :)
…
i) … Hint: How can you determine arc length?

ii) Prove A is perpendicular to B if |A+B| = |A-B|

iii) A||B if AxB=0

Ug! Welcome to PF!

i) arc-length = radius times angle (in radians)

(so arc-length for 2π angle is 2πr, as you'd expect! )​

ii) Hint: square each side!

iii) Your answer looks fine to me!

(I originally had 4 questions but solved one just by putting it into a different form of media) thanks a lot!

Yes, sometimes rewriting a question so that other people can understand it makes all the difference!

Ooog!

 

[thomate1]

I would like to commend you for your effort in attempting these questions. It is always important to ask questions and seek help when needed, especially when it comes to understanding mathematical concepts.

To answer your first question, to calculate the shortest distance between two points on a sphere (in this case, the Earth), you can use the formula for arc length: s = rΘ, where s is the arc length, r is the radius of the sphere, and Θ is the central angle (in radians). In this case, the radius would be the radius of the Earth, and the central angle can be calculated using the dot product of the two position vectors. The dot product gives you the cosine of the angle between the two vectors, and you can use inverse cosine (arccos) to find the angle.

For your second question, you are on the right track. To prove that A is perpendicular to B, you can use the Pythagorean theorem. If you square both sides of the given equation, you get (A+B)^2 = (A-B)^2. Expanding this and using the fact that A and B are perpendicular (meaning their dot product is 0), you can simplify the equation to show that A and B are indeed perpendicular.

For your third question, your reasoning is correct. If A and B are parallel, their cross product would be 0, which is what you showed in your proof. Therefore, A and B are parallel if and only if their cross product is 0.

I hope this helps and keep up the good work in your studies!