Vector calculus question

[tazza]

I'm having trouble understanding a concept on vectors... how do I find the shortest distance between two vectors? The question has come up quite a lot, such as on VCAA 07.

It says on the VCAA examiners report to find when r.v = o? Is this true for all cases? I have never learnt anything like this in class.

[dc302]

Could you please post a more specific example?

When the dot product = 0, it means the vectors are perpendicular to each other, which is associated with the notion of 'shortest distance'. For example, if you want to cross a street in the shortest distance, you would cross perpendicular to both sidewalks!

[dc302]

I just had a look at the question in 07. The reason you should do r.v is because r is the position vector of the aircraft in relation to the control tower, and it is asking you to find when it is closest to the control tower.

So this means the control tower is at the origin, and you know that the position vector is basically a vector going from the origin to the position of the aircraft (think of it as a line). You should also know that the velocity vector always gives direction (velocity is speed and direction). So think of v as a line tangent to the path of the aircraft at any time t. What you want is to find when the 'straight line' from the origin to the aircraft is perpendicular to the tangent of the path.

Give it a shot! A diagram may be helpful.

[tazza]

Oh ok, that makes sense, thank you. I haven't learnt that concept in class before, so I had no idea how to approach it.