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**Question:**Given the points A(-1,2,1) B(0,1,1) and C(7,-3,0)... (a) find an equation of the plane through A,B and C. (b) find the area of the triangle ABC. For (a) I thought that I had to set up a systems of equations to solve for the normal vector, but that doesn't seem to be working out. Any help at all would be lovely.

**Answers:**(a) The vector from A to B, which is (0-(-1), 1-2, 1-1) = (1, -2, 0), and the vector from A to C, which is (7-(-1), -3-2, 0-1) = (8, -5, -1), both lie in the plane. You can therefore find a normal vector to the plane by calculating the cross product of those two vectors. (I'll let you do that part.) Since I didn't figure out that cross product, I'll just call it (a, b, c). You now know that the equation of the plane is ax+by+cz=d, where a, b, and c are the components of the cross product you just found. To find d: 1. Substitute the coordinates of one of the three given points (point B would probably be the easiest) for x, y, and z in ax+by+cz=d, and... 2. Solve for d. (b) The magnitude of the cross product of (1, -2, 0) and (8, -5, -1) is the area of the parallelogram formed by those two vectors. The parallelogram consists of triangle ABC together with an identical copy of it, so the area of the triangle is half that of the parallelogram; that is to say, it's 1/2 times the magnitude of the cross product.

**Question:**To me questions like these force me to think outside the box. Our history and our universe is on one timeline. We wake up, live our day, then go to sleep. This is what we know and we discover all of what we know in this timeline about the world surrounding us and inside of us. I consider this timeline to be viewed in one dimension. Has anyone considered the possibility of more than one timeline on seperate axis' existing all around our timeline? i.e. Parallel dimensions. What is the best theories on parallel dimensions? Are gaps between timelines considered? Inter-timeline travel? -Properties concerning each timeline - A timeline (us) One time interval, Days are 24-hours - B timeline, objects could dissapear and reappear in one time interval, while living organisms exist in another time interval. -C timeline The universe is growing and collapsing at the same time.

**Answers:**Time is one dimension, one of the famous four: height, width, depth, time, using commonplace terms. I could also write (i, j, k, t); where i, j, and k are unit vectors (e.g., i dot i = 1) designating the three spatial dimensions. And t would be the fourth dimension. Time is real...it's not just the passage of events, like rising, showering, breakfasting, etc. Time passes even if no events take place. Time can be stretched out so that, for example, it would take 2 Earth seconds for 1 second to tick off on a very fast spaceship. Such a stretch is called dilation and this phenomenon demonstrates that time can be manipulated. That is a prime clue that time is real. If it were not real, we wouldn't be able to dilate it. This dilation can be used to travel into the future. For instance, if a star trekker in the above example traveled one year according to his clock on the spaceship and then returned to Earth, he would find Earth time had advanced two years. In other words, the spaceman would find himself one year into his future when he stepped out of the ship. As to "parallel dimensions" you are mixing concepts. In fact it's higher dimensions and parallel universes. [See source.] String theory posits up to 11 dimensions instead of the conventional four we know and love. One aspect of the theory suggests the other seven (all spatial) are simply curled up so tiny (1 Planck length = 10^-33 cm) that we can't see them. But strings, because they are also tiny, can see the extra dimensions and are constrained by them just as we are constrained by the four dimensions of our universe. One WAG of string theory is the parallel universe. Each universe is like a slice of bread in a mega universe loaf. Each slice is separated by 1 Planck length and 1 Planck time (which is also very tiny but I've forgotten the number). One SWAG resulting from the WAG is that two or more of the parallel universes collided and rebound. That change in momentum over time gave rise to the tremendous energy we call the Big Bang. Thus, there would be a BB in our universe as well as another BB in the universe that collided with us. And so, if we count the BB as t = 0, the timeline of the two BBs starts at the same time the two parallel universes. That is not to say the chains of events are identical...it's unlikely they are. But time, a real dimension, will be identical. There would be a gap between the timelines of the two parallel universes. The time length of that discontinuity would be 1 Planck time. And there would be a spatial gap of 1 Planck length between the two after they rebound from the collision. Both these gaps result because, theorectically, the 1 Planck time and the 1 Planck length are the smallest possible intervals in time and space. Unless the makeup of the two colliding universes was significantly different, the makeup of the two BBs ought to be about the same. So the uniform initial energies of both would go through the same evolutions and end up with the same kind of galaxies, planets, and energies. This suggests there might be living, intelligent beings living out their lives in the parallel universe...wondering if there is life out there.

**Question:**Find 3 planes which intersect at the line r=(4,-5,6) + t(2,0,-1) Help? Why are you considering lines parallel to the x axis?

**Answers:**The dot product of the directional vector v, of the given line and the normal vector n1, n2, and n3, of any plane that contains the given line is zero. v = <2, 0, -1> n1 = <1, 0, 2> n2 = <1, 1, 2> n3 = <1, 2, 2> Point P(4, -5, 6) is on the line, and therefore lies in all three desired planes. With the normal vector of the plane and a point in the plane we can write the equation of the plane. Remember, the normal vector is orthogonal to any vector that lies in the plane. And the dot product of orthogonal vectors is zero. Define R(x,y,z) to be an arbitrary point in the plane. Then vector PR lies in the plane. n1

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