Find B given distance and direction from vector A

[WinstonB]

Homework Statement

"The distance D between A and B is 10. The vector A (2,-1,4) points towards B in the direction given by C=(2,-5,3). Find B."

Homework Equations

The Attempt at a Solution

I have no idea how to even start

 

[Simon Bridge]

Start by clarifying the description - it looks a little confused ..
eg. (1) it has used "A" as both a point and a vector here ... and the vector A does not point to point A. (2) It says that vector A points towards B in the direction of another vector C that actually does not point in the same direction as A.

 

[WinstonB]

The question is stated exactly as I have typed it here; I find the way it's worded is confusing.

I've just updated to correct an error - the first component of the vector C should be "2" so it's the same as the first component of the vector A.

 

[Simon Bridge]

C still does not point in the same direction as vector A, and vector A does not point at point A - but at point B. You need to get this cleared up with the person who set the problem.

I can guess

$\overrightarrow{OA}=(2,-1,4)$ and $\overrightarrow{AB}$ has the direction of $\overrightarrow{OC}=(2,-5,3)$ but has a magnitude of D=10units.
You are being asked to find $\overrightarrow{OB}$ ... I'd do it head-to-tail myself.

 

[lendav_rott]

What does it mean to point towards a Point in the direction given by another vector in a x:y:z space

Does it mean that all of these components are on the same plane ?

 

[Simon Bridge]

What does it mean to point towards a Point in the direction given by another vector in a x:y:z space

Well, you live your normal life in a 3D World. When you see something interesting in the distance and you want to show your friend, it may be in the sky over a bit to the left some distance or something, so you point at it don't you? You extend your whole arm and one finger, you say "it's over that way, a couple of miles out" or something and you have pointed in 3D.

Your arm is a vector with the arrowhead at your finger.
You are telling your friend that the object of interest is in the direction of your arm but not actually at your fingertip. If he looked at you finger and said: "what? I don't see anything!" You'd say "no dummy: it's in the direction of my arm but a couple of miles away!"

I think you need to revise the basic concepts of vectors.

 

[lendav_rott]

Ahh, this is so contradicting.
So B is the star they're looking at, vector A is pointing towards the start of vector C and vector C is pointing toward the star?

 

[Simon Bridge]

Well ... it could be that you have directions to get to star B like this: travel to star A, then head in the direction of C until you have gone D light years... and there it is.
What is so contradicting about that?

It could be that the inhabitants of star A have send a radio message telling you about a cool object that is D lightyears in the direction of C from where they are. You don't have the budget to visit star A first but you want to go look at it ('cause it's cool) so you need to know which way to go from here.

We do this sort of thing all the time ... you are on a cruise ship and you are phoning family and you say stuff like "I can see the shore over to my left" ... you don't tell them where the shore is in relation to them do you? That would be silly!

If you want to find a star in the sky - it is usually easier to start by finding a bright star first. So you say something like "the star you want is a few degrees left and down from Arcturus". I mean, if you wanted to tell a novice where to find the Pleiades, you start out by saying something like, "first find Taurus..." wouldn't you?