Also, Capella is the brightest star in the constellation Auriga.

]]>Here we go.

I'll use bold letters for the vectors.

First choose an origin. Any point will work but as the question is about the line BQR I'll choose B.

And for a 2-D problem you need two base vectors. Any will do providing they are not parallel so I choose **BA** = **a** and **BC** = **c**

The position of every other point can be written as some combination of an amount in the **a** direction and an amount in the **c** direction.

P is the midpoint of BC so **BP** = 0.5**c**

And **AP** = **AB** + **BP** = -**a** + 0.5**c**

=> **AQ** = 0.5(**AP**) = -0.5**a** + 0.25**c**

=> **BQ** = **BA** + **AQ** = **a** -0.5**a** + 0.25**c** = 0.5**a** +0.25**c**

**AC** = **AB** + **BC** = -**a** + **c**

Now we know R lies on AC but we don't know where, so suppose it is fraction f, along from A (normally I'd use a Greek letter lambda here but it gets too complicated to try and show that)

=> **AR** = f( -**a** + **c**) = -f**a** + f**c**

=> **BR** = **BA** +**AR** = **a** -f**a** + f**c** = (1-f)**a** + f**c** ......(1)

WE also know that R is some way along BQ, let's say g along where g is a number over 1.

=> **BR** = g (0.5**a** +0.25**c**) = 0.5g**a** + 0.25g**c** .......(2)

We now have two ways to describe **BR** so they must be equal. In vectors, each component must be equal so we have a pair of simultaneous equations:

**a** components: 1-f = 0.5g

**c** components: f = 0.25g

adding gives 1 = 0.75g so g = 4/3 and so f = 1/3

In (1) gives **BR** = 2/3. **a** + 1/3. **c** and in (2) gives **BR** = 2/3. **a** + 1/3. **c**

I've worked out both ways to check for mistakes. I did find one, so these are the corrected values.

As g = 4/3, we know BQ:BR = 1:4/3 = 3:4 => 3.QR = BQ as required.

Bob

]]>i like math because it is a "language" which is internationally available.

]]>Do you know the ellipse? The parabola sits on the border between the ellipses and the hyperbolas.

An ellipse is a single closed curve and the hyperbola is always open and has two disjoint parts.

This page: http://www.mathsisfun.com/geometry/conic-sections.html tells about the conic sections.

Start with a double cone (that is a cone and another upside-down cone sitting on its vertex.

If you cut (make a section) with a horizontal plane the shape of the section is a circle.

Incline the plane a little and you get an ellipse. As there are many angles for the cut there are many different ellipses you can make in this way.

Tilt the plane beyond the angle of slope of the cone and you get the 'family' of hyperbolas. The planes cut the lower cone and the upper cone so you get two sections. Again there are many angles that will result in hyperbolas.

When the angle of the cut equals the angle of the slope of the cone you get a parabola. The cut can slice the lower (say) cone but will just never cut again on the top cone. This is why I say it sits at the border between the hyperbolas and the ellipses.

Copernicus discovered that the planets orbit the Sun in elliptical orbits. Newton showed mathematically why this is so. It is possible for a planet to orbit a star in a circular orbit but in practice the orbit will always be at least slightly elliptical.

Comets that return also have elliptical orbits. But it is possible for an object to enter the solar system and pass close to the Sun before swinging back out and never returning. Such an orbit is a hyperbola. Again is it possible for the orbit to be a parabola but it's not likely.

Search lights use reflecting mirrors that are parabolic. If the bulb is at the focus of the parabola, then the light beam will produced parallel rays. For the same reason astronomical reflecting telescopes have a parabolic surface so that all the light gathered is concentrated at the focus.

Bob

]]>I am unable to provide a better explanation.

]]>Here is a fun video about this.

]]>It is a widely held view that God is a very good mathematician, because the 'rules' of the Universe can be expressed as mathematical equations.

It is also a view in many religions that God does not intend life for humans to be easy. God expects us to have to struggle with things, but also to overcome difficulties and be successful.

Putting these ideas together, I think this means that maths will be a struggle to understand but God would like us to try and that we can succeed.

Whatever, you have come to the right place. Here you can get help with understanding maths.

Bob

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