Hello, everyone, welcome back to Anjali math class. YouTube, channel, first, subscribe, my channel, my channel for more videos. Today we are going to discuss the new chapter that is three-dimensional coordinates.

The weightage of this chapter in IPA exam is one two mark question from the chapter. So already we know what is two-dimensional coordinate axes. So what is two diagonal important axes? So this is x-axis.

And this is x dash. This is y-axis. And this is y dash. This is two-dimensional coordinates. Axis suppose here origin is always called origin point of integration of two lines is called origin that is or 0 0. Suppose if we draw one more line perpendicular to x right n, one more line perpendicular to this x y plane, this is z axis. And this is the dash.

So that becomes three-dimensional coordinates. So here, o, x, o, y, o, z or called positive directions of coordinate axis and o. X dash. O, y dash. O. Z, dash are called negative directions of coordinate axes here. The three mutually perpendicular lines x.

Dash o, x, y dash, o, y, z, dash, o, z or call a rectangular coordinate axis. O, x. O, y, o. Z are called positive directions of opponent axis and uh, o. X, dash. O, y, z. O, y dash. O. Z, dash are called negative directions of coordinate axes the coordinates of origin.

Here we are talking about three dimension. So origin also will take three-dimensional or zero. Zero. Zero. Suppose, if p of x y, z is a point in space. So I will show one model that is three-dimensional geometry model three-dimensional coordinate system model.

So. This is prepared by me see this is the three-dimensional coordinates' axis. So this is called x-axis.

This is horizontal line. This is y-axis. And this is z axis. These three are mutually perpendicular lines. So x here, x dash.

Here y back side, y dash. This is z and this also z dash. So this is the origin here.

Horizon. This is the origin. Suppose, this plane is found by x-axis and z axis. Therefore, it is the exact plane it is exact.

And this plane is found by y-axis and z axis. This is y z plane. This plane is. Found by x-axis and y-axis, this is x y plane.

Suppose if any point lies on x-axis, if any point lies on x-axis, the point in the form of x 0 0, same as 2 2 dimensional coordinate axes. If any point lies on x-axis that poison point in the form of x 0 0. Suppose, if any point y, and if any point less than y-axis that point in the form of 0, y 0 0, y 0. Suppose, if any point nice and z axis that point in the form of zero zero zero. If any point likes an axis x-axis, the point in the form of x is zero and y. Axis zero y is zero and ex is zero.

Z. You see the plane found by x-axis and zero x is called exact pen in exact plane, the y coordinates of zero that is y is equals to zero. So we will take this equation of x. Z plane is y equals zero equation of x. Z.

Minus y is equal to zero c is here equation of y. Z plane is x sorry, equation of x. Z mean, is y is equals to 0 and see here. This plane is formed by y-axis and z axis. So this is called y z plane in y, z plane, x coordinate is 0 x equals. To 0, therefore, the equation of y, z, print is x equals 0 like. So this plane is formed by x-axis and y-axis.

So again, x y plane, z, coordinate is equal to 0, therefore that we will take. So the equation of x sub n is z equals zero because here, whey, z, coordinate will become zero. Therefore, we will take equation of extra places it equals zero. And the equation of y z plane is x equals 0. And equation of this explain is y is equal to 0.

Suppose, if any point p lies in space, space means equal space. The x. Coordinate from x-axis, only that is x coordinate and taken y coordinate y, I y pointer z. So x is called x coordinate x is if a p point x, y, z is a point expressed. Then x is called the x coordinate y is from the y coordinate and z is for the z coordinate next point. The three coordinate planes divide the space into eight equal parts called each part is called octane. The octane found by the edges.

O, x. O. Y o. Z is called the first order. And it is divided by o. X, y, z, see this our model of three.

Dimensional coordinate axis. So this is o x. This is o. Y, this is o. Z. So total x, y x, total planes, divide into space into eight equal parts. Each part is called octane. This is first octane.

This is second octane. This is third octane. And this is fourth octane and bottom also we have total force, uh, four, octanes, top low foreign is found by tax yo. Z.

Therefore, this oxygen is called o. X, y. Z. Octane. Second, octane is found by o. X, dash, o, y, o. Z. So this is called o. X. Dash. Y.

Z third is found by o. X dash. O, y dash. O z, so this is third accent. Fourth is the extreme backside. Fourth.

And so that is fourth quadrant. Suppose already we know like two-dimensional geometry, we know how space the plane divide into my x, y, x, x and y-axis. So this is first quadrant. This is second quadrant. This is third quadrant. And this is fourth quadrant.

And then a point in the quadrant, one then quantity then sign around on the sign. So under first quadrant law, both x-axis and x coordinate and y coordinates are both are positive in second. And second quadrant index is negative and y is positive. Third quadrant, both are negative and fourth quadrant of x, positive, y negative.

So this is a science of points whether the line in first quadrant or second quadrant or third and fourth depend on quadrant. You can go the same as point underneath the total eight octanes. This is coordinated three-dimensional coded axes. Three-dimensional coordinate axis is the o x o, x and a positive o, x, plus and o. Positive x-axis. Okay. Now.

So this is first y-axis. This is positive, z axis, even the positive way. So first quadrant together first quadrant of all are positive integrity, our x coordinate positive, y coordinate positive and z, coordinate positive. So x square, first quadrant, x, coordinate positive.

Second, quadrant, uh, third order, z, coordinate also positive. Second, x, simple, x, kind of positive, whether back side and move to the fourth quadrant then backside. And the fourth executive first club position, then backstage when you pass through x-axis together. So.

Then backstage would have passed you into the and the first level positive and four through past two. Next then bottom row. Bottom left fifth quadrant into another fifth grade quadrant found by the o x o, x and o. Z, dash. So o, x and take care of fifth lag, x pass. Twenty then back sided in the eighth quadrant under the and x negative. And a second length of y curve, y positive together then catastrophe.

Second quadrant together and the second y positive, z, also, positive level, our simple. Next third quadrant, third, quadrant, x, dash, backside, a round exposed towards the y negative also is that positive. Also in the first four quadrants first forward current is the first two x-axis is the first pass one to the first quadrant and fourth row, the positive. Why the same second only a second lower sorry, second, first, first to second, positive. So first positive, why is this attaching the second together positive exercising? Then cut out the second chord?

Second octane, there second loop, positive, Remaining backside, three, four negative, same method. Why enter my fifth position to the sixth positive seventh eighth negative said, topless, positive is that access together. And a total top four upgrades are found with the positive z axis. And the top four, the total x is positive. Z is positive. And then next to fifth quadrant is if the fifth column dot XML is positive.

Next y one, two, three, four positive, z and negative zero negative and sixth row, x negative, y positive, z, negative seventh row. Negative. Negative negative, eighth row falls to negative, the data science. So making monarchy 2d damage, stay down, two dimensions by positive. First, two negative, negative, positive, positive, negative.

Again, one, two three four rather totally attacked and said that five six seven, eighty only positive negative. Zero, top four, top one to the top four. Okay. These are very, very important point, uh, important for x-axis and y-axis that's. Imagine x and positive x-axis, positive, y. Axis, positive, z axis, a one two, three, four, five, six, seven eight. So x could pose to one and four five and eight. Why?

One two, uh, one, two, five, six remain low negative. Zero, top four, top four under one two, three, four pulse, two next bottom, four, five, six, seven, eight, zero negative. Okay. Now, the significant formulas, we use in this chapter.

First formula, distance formula. Already, we know, uh in two-dimensional geometry. What is the distance between x 1?

Y 1 and x 2 by 2, x 2, minus x 1, whole square, plus y 2, minus 1. Over entire square under root can extension z1 and q of x2 y2 z2 or any points then distance between p and q is x2 minus x1 whole square, plus y2, minus y1 whole square, plus z2, minus z1 whole square under root distance between p and q. The if an of x1 was z1 b of x2, y2, z2, c of x3, y3 z3 are collinear. Then ab plus bc is equal to ac and ac plus b. C b is equals to a b or b, a plus a c is equals to b c and a b, b, c, c, a financially of any two sides of the outside travel, definitely they need to put line. Segment chart is a no section formula.

If p of x, y x, y, z, divide, the line segment joining the points, an of x 1 and z 1 and b of x 2, y 2, z 2 in the ratio m is to n is coordinates of p is equals to m, x 2, plus n. X 1 by m, plus n, comma, m by 2, plus n, y 1 by m, plus n, comma, m, z 2, plus n. Z 1 by m, plus n. This is our line segment joining a. This is a point you could a point an it cannot be pointed. So this is p pointed. So coordinates of p is equals to x y.

Z. This p point is equal to the divides a b in the ratio, m. Is t, and this is a p is equal to m. And p b is equals to n the m1 calling the m. So coordinates of p is equal to this the which divides a b in the ratio of that point is equals to m, x 2, plus n. X 1 by m, plus n. Comma, m, y 2, plus n. Y 1 by m, plus n, comma, m, z, two, plus n. Z, one by m, plus n, the midpoint of line segment joining the points, an of x one, one, z, one and b of x, two, y, two. Z. Two is same with the formula midpoint. And take p, I kinda subscribe, my plus one into y. One y one plus y two by two x. Zero, plus z, two by two.

So this is a midpoint of a and b, two points is a middle point in t, x, one, plus x, two by two y, one, plus y, two by two comma zero, plus z. Two by two y, z plane, divides the line segment joining points, an of x one r, one, z, one and b of x, two, y, z, two in the ratio, minus x 1 is to x 2 already, my two dimensions, the center of a triangle, whose vertices are an of x 1 and z 1 and b of x 2, y 2, z 2 and c of x 3, y 3. Z, 3 is x 1, plus x 2, plus x, 3 by 3, y 1, plus y 2, plus y 3 by 3 comma, z 1, plus z 2. Plus z, 3 by 3. This is a centroid of a triangle. Okay.

Now tetrahedron that is a shape of a tetrahedron. It has four vertices tetrahedral, nothing, but triangular pyramid triangular, permanent, hitter, hetero atom. So it has four vertices and six edges.

The tetrahedron is a closed figure found by the four planes or not all passing through the same point. See that figure all planes not passing to the same point. It has four vertices and six edges, the centroid of a tetrahedron divides. The line. Segment in the ratio three is to one each line segment divides, uh by divides by tetrahedral in the ratio three is to one.

Okay. Triangular two is one ratio one to the tetrahedral centroid, divides, uh, each line segment in the ratio two is twenty tetrahedra. Low 3 is to 1 it divides with the ratio of 3 is to 1.

These are you the centroid of a tetrahedron whose vertices are an of x 1, y 1, z 1, b of x 2, y 2, z 2 and c of x 3, y 3, z, 3 and d of x, 4 y, 4. Z. 4 is x 1, plus x 2, plus x, 3, plus x, 4 by 4 y 1, plus y 2. Plus y, 3, plus y 4 by 4 z 1, plus z 2, plus f, 3 plus z, 4 by 4 these two exhale or remaining formulas will be lowest. And that is centroid of triangle center head of tetrahedron and midpoint Vera a second acceleration.

Dated : 19-Mar-2022