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Let us try to follow the numbers as an equation is re-arranged.

The slope is -a/b

Any line parallel must have the same slope, -a/b

You could do this:

Example Find a line parallel to 3x + 5y - 7 = 0

The slope is -3/5 so I'll make up a new line with the same slope:

Please notice that the ratio 3:5 appears all the way through this problem. Once you realise this will always happen, you don't need to make the slope intercept form to answer the question. It is easy to write answers down just by using 3 and 5.

Like this. Here are some more possible answers:

3x + 5y -12 = 0

3x + 5y + 100 = 0

6x + 10y + 20 = 0 [because this is the same as 3x + 5y + 10 = 0]

300x + 500y + 10000 = 0

.......................

.......................

Bob

hi Zeeshan 01

In that line the 1 in 1x and the 3 in 3y completely determine the slope of the line. The -9 only affects where the line crosses the axes.

As a parallel line must have the same slope, any of the following will be parallel:

x + 3y + 10 = 0

x + 3y +100 = 0

x + 3y = 97

even

2x + 6y + 17 = 0 because the ratio is still 2:6 = 1:3

So these will also work:

2x + 6y + 100 = 0

12x + 36y - 1000 = 0

even

0.5x + 1.5y + 1.23456 = 0

You just have to make sure that in the equation ax + by + c = 0, a:b = 1:3

So, there plenty to choose from (an infinite number as there's an infinite number of parallel lines). Just make sure you don't accidentally pick the same line as the question:

eg. 2x + 6y - 18 = 0 is just the question line doubled.

Please make up one of your own rather than pick one of mine. That will show you have understood.

Bob

Yes, I agree. That's why I put 'excuse' in inverted commas.

There's another reason too. In three year's time, when she has to have an election, the Brexit deal will be known by all. She may not be so popular then.

But didn't she argue that the Scottish people shouldn't be allowed a referendum until the facts are known. That's funny; it's almost as if she makes up the rules to suit herself.

Bob

Yes. They are on the same horizontal line.

Bob

Yes. That's the formula I was referring to.

note the bracket:

D=(ax1+by1+c)/sqrt (a^2+b^2)

Bob

With the combination of global warming and Brexit, I expect we'll be growing our own bananas soon here in the UK.

Theresa May's 'excuse' for calling an early general election: She's fed up with the opposition MPs actually trying to prevent her from doing what she wants. She's hoping there'll be no opposition at all after June 8th. She may (no pun intended) get her wish.

Before the referendum she was a Remain supporter. Her election constituency voted in favour of Remain in the referendum. Odd that she is now a Brexit supporter.

Bob

hi Agnishom,

An oft-cited objection to Britain being a member of the European Union is that the EU forces silly 'directives' on member states. The bent banana directive has been used for years by Eurosceptics as a reason that belonging is foolish. The Wiki article explains quite well the true facts.

https://en.wikipedia.org/wiki/Commissio … nt_bananas

Just in case you think British citizens can only buy straight bananas here is a photo of the shelves at a well known supermarket not far from my house:

During the 'Brexit' debate a lady (in the audience) said on national television that she had decided to change her vote from 'Remain' to 'Leave' because she wanted to be able to buy bent bananas. She was widely ridiculed but the Daily Mail continued to run the directive criticism story.

The Leave campaign had a battlebus:

The financial editor of the BBC demonstrated on Tv that this figure was completely untrue.

The (unbiased) head of the UK Statistics Authority, Sir Andrew Dilnot, censured the £350m a week claim and told the Leave campaign they should alter it. They were also widely criticised in the media.

They changed to 'poster' to read '50 million a day' !

The day after the referendum result Nigel Farage admitted that it was a lie.

Sometimes I despair of there ever being intelligent debate in this country. sob sob!

Bob

If you want to do this using that formula you need (x1,y1) to be (any) point on one line, and (x2,y2) to be a nearest point to it on the other line. You could proceed like this:

step 1: choose a point on line 1 ; A = (x1,y1)

step 2: calculate the gradient of the line through A perpendicular to line 1.

step 3: calculate the equation of the perpendicular line (say, line P)

step 4: calculate where line 2 crosses line P ; B = (x2,y2)

step 5: calculate the length of AB.

I think it is quicker to calculate the difference between the intercepts = D. Then the angle of slope of the lines, θ. Then the required distance is D.cos(θ)

There is also a formula for calculating the distance from a known point to a known line. Ask again if you want this formula.

Bob

hi Zeeshan 01

The scalar product is defined that way. It is useful because the cosine of a vector gives the component of that vector in a certain direction. So the scalar product can be used to determine

eg1. Magnetic flux is the dot product of the magnetic field and the area vectors.

eg2. Work done by a force is F.d where the Force is F, and d is the vector for the distance moved.

Bob

I have edited my previous diagram.

Bob

Your sign is wrong in 2x?

Maybe this was deliberate so that you could work your question on your own rather than just copy a model answer. Anyway, why does it matter? You wanted to be shown the method and you have been shown.

Bob

It is the same formula. tanith has re-arranged it to make the angle the subject to make it easier for you.

Note also: it is called the 'dot' product because a 'dot' is used rather than a times sign. It's best to stick to this because {vector 1} x {vector 2} means use the 'vector' product which is a very different thing.

Bob

First question What have you got for the change in KE?

next question: Looks like the first with values for the velocities.

he only takes into consideration the i direction which does not make sense to me

for m you have shown that W = 0 so there is no change of KE in the y direction ( zero to zero)

For Km you know C = B so again no change in KE in the y direction.

Bob

hi Zeeshan 01

You can make an equation for this using the 'scalar' or 'dot' product. Do you what a scalar product is?

Bob

Two ways to say the same:

a < b + c is the same as b + c > a

a is shorter than b + c is the same as b + c is greater than a.

Bob

Initially you need 8 components for a problem of this type. You are given that two are zero, but must prove that some others are also zero. I suggest

mass km m

before Ai Xi

Bj Wj

after Yi 0

Cj 0

Your first equation is correct.

The second is the wrong way up. Final relative velocity above and initial relative velocity below.

From these two you can show that A = 0

Third is Bkm + Wm = Ckm

fourth is B - W = C

From these you can show that W = 0 and B = C

A = W = 0 shows that the initial velocities are perpendicular.

B = C shows that the j component velocity for km is unchanged by the impact.

Now you can calculate the changes in KE.

Bob

Or is this from the topic of vectors?

Bob

hi markosheehan

The coefficient of restitution is taken to be a positive number between 0 and 1 (inclusive). For an object that bounces off a fixed surface, clearly it will change direction after the impact. For a pair of objects colliding and bouncing apart it is necessary to consider the relative velocities. The law, usually attributed to Newton, incorporates a negative sign in order to cover this point:

Note that, in the question, positivity is implied:

the coefficient of restitution for the collision is 1/k (k greater or equal to 1

I have successfully done all parts of this question. I started with initial velocities for each as two components in the direction made by the centres (x) and perpendicular (y). Similarly two components after the collision.

I constructed four equations: (1) conservation of momentum in the x direction; (2) same in the y direction; (3) restitution law in the x direction; (4) no change in relative velocity in the y direction.

I don't think you will succeed with less.

Bob

hi kapitanamerika

Welcome to the forum.

I suspect a catch question. You have 12 matches. I notice that you can make the four letters of NINE (a square number) using 11 matches. One snag: I need to move 4 matches not 3.

Someone will come up with a better solution I'm sure.

Bob

I made a drawing using 'Sketchpad', and got about 3.6

That's not to say I'm disagreeing with your answer. I may not have been sufficiently accurate. Please post your method.

Bob

hi Mystogan

That series is a geometric progression so you can use the formula:

http://www.mathsisfun.com/algebra/seque … etric.html

Bob

Apparently, but really there's only two: the 'in it for how much I can make' party and the 'I really want to make the world a better place' party, which is everyone else.

Unfortunately, the media tend to promote the former.

Bob

If the centre is point H, you can enter all the angles around H and hence get angle HDG in an isosceles triangle. So splitting it in half gives you a 90 angle for some trig.

Bob