First year maths questions (plz elp) (1 Viewer)

[RG11]

1. If A(-1,3,4), B(4,6,3), C(-1,2,1) and D are the verticies of a paralellogram, find all the possible coordinates for the point D.

2. Consider three non-collinear points D, E, F in R^3 with coordinate vectors d, e and f. There are exactly 3 points in R^3 which, taken one at a time with D, E, and F, form a paralellogram. Calculate vector expression for the three points.

3. Construct a cube in R^3 with the length of each edge 1. Show that the face diagonal has length sqrt(2) and the long diagonal sqrt(3). Try to generalise this idea to R^4 and show that there are now diagonals of length sqrt(2), sqrt(3) and 2. How many verticies does a 4-cube have?


4. Find the cosines of the internal angles of the triangle whose verticies have the coordinate vectors A<4,0,2>, B<6,2,1> and C<5,1,6>.





I know theres quite a few questions but ive been stuck on these for a while and would be very grateful for some help

 

[Elise8842]

Is it maths 1151?


~

 

[RG11]

nope 1131 lol

 

[InteGrand]

1. If A(-1,3,4), B(4,6,3), C(-1,2,1) and D are the verticies of a paralellogram, find all the possible coordinates for the point D.

2. Consider three non-collinear points D, E, F in R^3 with coordinate vectors d, e and f. There are exactly 3 points in R^3 which, taken one at a time with D, E, and F, form a paralellogram. Calculate vector expression for the three points.

3. Construct a cube in R^3 with the length of each edge 1. Show that the face diagonal has length sqrt(2) and the long diagonal sqrt(3). Try to generalise this idea to R^4 and show that there are now diagonals of length sqrt(2), sqrt(3) and 2. How many verticies does a 4-cube have?

4. Find the parametric vector forms to describe the following planes in R^3.
a) x2+6x3 = -1
b) x3=2

5. Find the cosines of the internal angles of the triangle whose verticies have the coordinate vectors A<4,0,2>, B<6,2,1> and C<5,1,6>.





I know theres quite a few questions but ive been stuck on these for a while and would be very grateful for some help

1. Let .

There are three possible locations for the point D, namely we can have the following as parallelograms: ABCD, DACB, or ABDC.

I'll show how to find the point D for one of these, and the others are similar.

For the parallelogram to be DACB, we require (opposite sides of parallelogram are equal and parallel).

So .

Equating the components of the vectors yields .

(Draw a diagram to see that there are indeed three places where D could go to get a parallelogram.)

2. We use the same method as above. Let the last required point be X, and note that there are three possibilities for X, as we can have the following three parallelograms: DEFX, XDFE, DEXF. We show how to find one of the possible X positions (the other ones are found similarly).

To make the parallelogram DEFX, we require (opposite sides of parallelogram are equal and parallel).

i.e. we need .

For the other two X positions, just change the position of the minus sign above to the other two possibilities (i.e. in front of f, or in front of d) and have the other two signs and plus signs.

3. A face of the cube ABCD can be described by these points: A(0,0,0), B(0,1,0), C(0,1,1), D(0,0,1).

The diagonal of the face is .

The vertex of the cube opposite A is at (1,1,1), so the length of the long diagonal is .

For a 4-cube, we'd have the diagonal of a square (a 2-cube) and the diagonal of 3-cube "within" our 4-cube, and these have lengths and . For the longest diagonal, it's the distance between the points (0,0,0,0) and (1,1,1,1), which is .

4. a) Set and , then rearrange the plane equation to find in terms of and . Then write and substitute the expressions for the in terms of and separate out to get the scalars outside the vectors.

b) Similar process to a).

5. Use the formula .