Vector Equations and Circle Problems

[pavadrin]

Hey
There are a few problems regarding vectors which I am not sure on how to go about solving them. These problems are listed below along with my working.

_____________________________________​

Find the vector equation of the circle centre C, position vector -7i+4j, and a radius of $$4\sqrt{5}$$ units. Determine wether the point A, position vector i+8j, lies inside, on or outside the circle.
My working:
Vector equation:
$$\mid{r - (-7i + 4j)}\mid = 4\sqrt{5}$$
$$\mid{r + 7i - 4j}\mid=4\sqrt{5}$$
As for the second part I have used the vector equation and substituted r with vector a
$$\mid{(i+8j)+(7i-4j)}\mid = 4\sqrt{5}$$
$$\mid{8i+5j}\mid <> 4\sqrt{5}$$
(<> means does not equal, I don’t know the latex code)
However on the answers it says that this point does lie on the circle, so I know I’m wrong but don’t know where or why.

_____________________________________​

Find the distance between the centres of the two circles given below
$$\mid{r - (3i+7j)}\mid = 6$$
$$\mid{r-2i - 9j}\mid = 7$$
As for this problem I don’t know where to start.

_____________________________________​

The circle $$\mid{r – (3i – 4j)\mid =5$$ has the centre A and $$\mid{r – (2i + 7j)}\mid = 3$$ has the centre B. Find the euation of the straight line A and B.
Again this is another problem which I’m unaware on how to approach

_____________________________________​

thank you to people who post a reply, and even hints are appreciated for the problems which I do not know how to solve
Pavadrin

 

[Tide]

HINT: 8 - 4 = 4

As for the distance between the centers of the circles you must first write vectors (properly!) for the centers.

 

[pavadrin]

HINT: 8 - 4 = 4

As for the distance between the centers of the circles you must first write vectors (properly!) for the centers.

i don't understand ur hint
The correct vector eqyutions for the circle centres are
$$\mid{r - (3i + 7j)}\mid = 6$$ and
$$\mid{r - (2i - 9j)}\mid = 7$$

 

[rhj23]

the hint is in reference to the final step in your workings for the first question.

also, those are not the correct circle equations for the second. multiply out the lower one to see.

 

[pavadrin]

okay i get the first question now, i made a stupid mistake

 

[pavadrin]

as for the other question, I am still uncertain on what i have to do

 

[arunbg]

All you need to do with the second and third questions is to find the centres of the circles . Are you familiar with the vector equation of a circle ?
In the question 'r' stands for the position vector of any general point on the circle . What can you say about the remaining quantity in the modulus ?

By the way, in 3D these equations are technically those of a sphere .

 

[pavadrin]

All you need to do with the second and third questions is to find the centres of the circles . Are you familiar with the vector equation of a circle ?
In the question 'r' stands for the position vector of any general point on the circle . What can you say about the remaining quantity in the modulus ?

By the way, in 3D these equations are technically those of a sphere .

I'm not quite farmiliar with vector equation of a circle as this is a new topic which i have only recently started, so i know for sure these aren't 3D vectors/equations. However I do know that for a circle with the centre (0,0) the equation is simply $$\mid{r}\mid = a$$ where a is the radius. Therefore does this mean for a vector equation i.e., $$\mid{r - 2i + 4j}\mid = a$$ that $$- 2i + 4j$$ is the circle centre? thanks

 

[HallsofIvy]

No. If r is the position vector of a point on the circle and c is the postion vector of the center of the circle, then the distance between them, |r-c|, is a constant: |r- c|= a, not |r+ c|= a. If the equation is
|r- 2i+ 4j|= |r- (2i- 4j)|= a then the center of the circle is 2i- 4j.

 

[pavadrin]

Thank you HallsofIvy

_____________________________________​

Okay, now going back to the second problem on my first post:

Find the distance between the centres of the two circles given below
$$\mid{r - (3i+7j)}\mid = 6$$
$$\mid{r-2i - 9j}\mid = 7$$
As for this problem I don’t know where to start.

I have calculated the distance between the circle centres as being i-16j, or $$\mid{sqrt{257}}\mid$$. They way which I completed this calculation is shown below.
$$\mid{r – (3i+7j)}\mid = 6 \therefore circle centre = 3i+7j$$
$$\mid{r – (2i-9j)}\mid = 7 \therefore circle centre = 2i-9j$$
$$AB=AO+OB$$
$$AB=-OA+ob$$
$$AB=-(3i+7j) + (2i-9j)$$
$$AB=i-16j$$
I’m not sure on what to do about the radius of the circle, thus in this calculation I have simply ignored it. Is this correct? thanks

 

[arunbg]

Check your algebra.
AB = -i - 16j .
Otherwise the soln. looks fine.
Also remember that distance is always a +ve quantity, not a vector .
The radii are of no particular use in this problem.

 

[pavadrin]

ooooh i c, thanks arunbq