Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes


Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  Here we provides you Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes in PDF Format. Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes topics are direction cosines and direction ratios of a line joining two points.Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines.Cartesian and vector equation of a plane.Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.

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Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry : In mathematics, analytic geometry (also called Cartesian geometry) describes every point in threedimensional space by means of three coordinates.Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  In this chapter we present a vector–algebra approach to three–dimensional geometry. The aim is to present standard properties of lines and planes, with minimum use of complicated three–dimensional diagrams such as those involving similar triangles. We summarize the chapter: Points are defined as ordered triples of real numbers and the distance between points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) .

By taking a look at parallelism and perpendicularity in three dimensions, we see that many of the same principles apply but that several complexities are also added.

Key Terms

o Skew lines

o Axes

o Cross section

Objectives

o Recognize perpendicular, parallel, and skew lines in three dimensions

o Be able to represent simple three-dimensional figures using two-dimensional drawings

o Know what a cross section is and how it can be used to relate two- and three-dimensional geometry

Objects in real-life have three spatial dimensions, so it is helpful for us to understand some of the fundamental aspects of three-dimensional geometry. In many ways, three-dimensional geometry is just an extension of two-dimensional geometry; many of the same principles apply. In this article, we consider parallelism and perpendicularity in three dimensions, and we then take a look at drawing three-dimensional figures on a two-dimensional surface. Finally, we look briefly at cross sections.

Parallelism and Perpendicularity in Three Dimensions

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  In two dimensions, we noted that any two lines that do not intersect are parallel. Recall, however, that we gave a slightly more fundamental definition that involved the relationship of the angles formed by a transversal that cuts across the parallel lines. Now, imagine two lines in three dimensions. In three dimensions, you can probably easily imagine two lines that are not really parallel but that do not intersect, either. An example is shown below using lines l and m, where one of the lines (l) is shown “on end” (that is, going into and out of the surface of the page) so that it appears as a point.

Notice that the two lines do not have any point of intersection, and yet, they are not what we would consider “parallel.” These are what we call skew lines. Note also that these lines, although they look in some sense like they are “perpendicular,” are not truly so because they do not intersect. If they did intersect, however, then they would indeed be perpendicular, as shown below.

In two-dimensional (planar) geometry, for a given point on a line, there exists only one perpendicular line, as shown below.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  In three-dimensional geometry, there exist an infinite number of lines perpendicular to a given line. Consider a line l that intersects a plane at a right angle (in other words, wherever an angle measurement is taken around the line with respect to the plane, it is always 90°). We can draw innumerable lines in the plane that intersect line l; because they lie in the plane, they intersect l at a right angle.

Likewise, in planar geometry, there exist only two lines that are parallel to a given line for some fixed distance d, as shown below.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  In three-dimensional geometry, once again, there exist an infinite number of lines parallel to a given line for some fixed distance d. Imagine the line l passing at a right angle through a plane once again, as shown above. This time, however, let’s draw a circle with radius d in the plane.

Now, we can draw any number of lines that intersect the circle and the plane at a right angle-these lines are also parallel to l.

We can define parallelism and perpendicularity for planes in three dimensions similarly to the way we defined them for lines in two dimensions. Any two intersecting planes (that are not the same plane) form a line of intersection; this is the three-dimensional analog of a point of intersection for two lines. Such an intersection of two planes is shown below.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  If two planes do not intersect, then they are parallel. If the angle of intersection is 90°, then the planes are perpendicular. This angle of intersection is measured by looking at the planes “on edge,” such that the line of intersection is perpendicular to the surface of this page, as shown below. (Notice that when looking at the planes from this perspective, they appear to be lines, and the line of intersection appears to be a point.)

Want to learn more? Take an online course in Geometry.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry :  Thus, we can see that parallelism and perpendicularity in three dimensions, although they bear resemblance to the same concepts in two dimensions, take on a slightly more complicated character. Nevertheless, by extending what we have already learned in two dimensions, we can understand geometric concepts in three dimensions as well.

One might wonder if geometry can be extended to four or more dimensions. The answer is yes, but such an endeavor becomes much more difficult because we do not have any experience in reality that helps us conceive of a geometric space with more than three dimensions. For instance, although we readily understand the concepts of up-down, left-right, and forward-backward (the three spatial dimensions), we have a very difficult time conceiving a fourth dimension. What would such a dimension (or direction) look like? Such multidimensional geometries do play an important role in certain fields of study, but they are not nearly as applicable to everyday life as two- and three-dimensional geometry.

Practice Problem: Show that two lines that are cut by a transversal, thereby forming congruent corresponding angles, are not necessarily parallel in three-dimensional geometry.

Solution: This problem requires us to apply some critical thinking regarding lines and planes in three dimensions. First, let’s start with two lines that we will assume are parallel, and let’s cut these lines by a perpendicular transversal, as shown below.

Now, consider the transversal line, n. We know that in three dimensions, there exist an infinite number of lines perpendicular to n for a given point of intersection; these perpendicular lines all lie in a plane. Let’s draw the plane for the points at which parallel lines l and m intersect n.

But what if we “rotate” either line l or m in its respective plane of perpendicularity? For one, we know that lines l and m would no longer be perpendicular; they would become skew lines. Furthermore, we know that the angles formed by the intersection of l and n (as well as m and n) would remain the same, since these pairs of liens would still be perpendicular, as shown below.

Thus, even though corresponding angles for lines l and m cut by transversal n are congruent (they are all 90°), the lines are not necessarily parallel.

Drawing in Three Dimensions

Obviously, it is difficult to draw a figure in three dimensions when one is using a two-dimensional sheet of paper (or computer screen). Nevertheless, we can very easily make drawings that look three-dimensional by following a few simple rules. First, define directions (rays) for each dimension. Although it is not necessary, it is sometimes helpful to draw a set of axes, such as those shown below.

Note that we have shown that the axes are all mutually perpendicular-just as the directions left-right, up-down, and backward-forward are all perpendicular. To draw a figure that has a three-dimensional appearance, use distances parallel to a particular axis to create the illusion that a figure is extended in that direction. Note, for instance, how a cube shows three-dimensional character by virtue of its relationship with the axes (even if you ignore the axes, the same effect is achieved).

Practice Problem: Use a set of axes to draw a triangle that appears to be “leaning over” in three dimensions. (In other words, the plane formed by the triangle is not parallel with the surface of this page.)

Solution: Start by drawing a set of axes, as we did in earlier.

Now, let’s pick three points for the triangle, and let’s place one at some point on each axis. This will create the effect of the triangle having extent in three different directions (dimensions). Finally, connect the points, as shown below.

Thus, we have created a triangle that appears to be “leaning over” in three dimensions. To be sure, the triangle is still just a two-dimensional drawing, but by using the axes, we can create the illusion (albeit in a somewhat simplistic manner) that the triangle is “slanted” relative to the surface of the page.

Cross Sections

Consider some three dimensional figure, such as a cylinder; next, imagine a plane that cuts through that figure, as shown in the example below. The intersection of the plane and the three-dimensional figure is called a cross section, and it is a two-dimensional “slice” (of infinitesimally small width) of the figure.

In the example above, the plane of the cross section is perpendicular to the vertical orientation of the cylinder-the cross section is thus a circle. If use a different angle, however, we end up with an oval instead of a circle for the cross section.

Thus, we can see that two-dimensional geometry can in some sense be seen as a “cross section” of three-dimensional geometry.

Practice Problem: What two-dimensional figures are formed by taking the cross section of a sphere (a racquetball, for instance, is an example of a sphere)?

Solution: Recall that a cross section is simply a planar “cut” of the figure. Thus, let’s draw a plane and a sphere.

Now, a cross section is formed by lowering the sphere (in this case) into the plane, just as we might lower it into a pool of water. Because the sphere is perfectly symmetric, the resulting cross section is always a circle. Depending on where the cross section is taken relative to the center of the sphere, the circle may have a radius ranging from zero (the plane is tangent to the sphere, or just touches it at a point on the surface of the sphere) to the radius of the sphere.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry : The topic of three dimensional geometry is quite important and a bit complicated as compared to its counterpart two dimensional geometry. One of the ways of describing a 3D object is by approximating or assuming its shape as a mesh of triangles. A triangle is generally defined by three vertices wherein the positions of the vertices are described by the coordinates x, y and z.

The major heads that are included in 3D coordinate geometry are the direction ratios and direction cosines of a line segment.

Various Topics Covered in this Chapter Include: 

  • Rectangular Co-ordinate System in Space
  • Direction Cosines of Line
  • Direction Ratios
  • Parallel Lines
  • Projection of Line
  • Theory of 3D Plane
  • Theory of 3D Straight Line
  • Shortest Distance-two Non Intersecting Lines
  • Theory of Sphere
  • Solved Problems of 3D Geometry

What do We mean by Direction Cosines of a Line Segment?

The direction cosines are the cosines of the angles between a line and the coordinate axis. If we have a vector (a, b, c) in three dimensional space, then the direction cosines of the vector are defined as

cos α = a/ √(a2 + b2 + c2)

cos β = b/ √(a2 + b2 + c2)

cos γ = c/ √(a2 + b2 + c2)

http://cdn1.askiitians.com/Images/201423-134057832-5239-direc.gif

What are Direction Ratios?

If l, m and n are the direction cosines then the direction ratios say a, b and c are given by

l = ± a/√ Σa2

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m = ± b/√ Σa2
n = ± c/√ Σa2

Some Key Points:

  • While the direction cosines of a line segment are always unique, the direction ratios are never unique and in fact they can be infinite in number.
  • If the direction cosines of a line are l, m and n then they satisfy the relation l2 + m2 + n2 = 1.
  • If the direction cosines of a line segment AB are (l, m, n) then those of line BA will be (-l, -m, -n).

Angle Between Two Lines

Let us assume that θ is the angle between the two lines say AB and AC whose direction cosines are l1, m1 and n1 and l2, m2 and n2 then

cos θ = l1l2 + m1m2 + n1n2

Also if the direction ratios of two lines a1, b1 and c1 and a2, b2 and c2 then the angle between two lines is given by

cos θ = (a1a2 + b1b2 + c1c2)/ √(  a12 + b12 + c12) . √ (a22 + b22 + c22)

What is the Condition for Parallel or Perpendicular Lines?

When the two lines are perpendicular, the angle between the lines is 90° which gives the condition of perpendicularity as

l1l2 + m1m2 + n1n2 = 0

or this implies a1a2 + b1b2 + c1c2 = 0.

Similarly, when the two lines are parallel, the angle between them i.e. θ = 0.

This gives l1/l2 = m1/m2 = n1/n2

This also gives a1/a2 = b1/b2 = c1/c2

What is the projection of a line segment on a given line?

Suppose we have a line segment joining the points P (x1, y1, z1) and Q(x2, y2, z2), then the projection of this line on another line having direction cosines as l, m, n is AB =  l(x2-x1) + m + m(y2-y1) + n(z2-z1).

What Exactly is a Sphere?

A sphere is basically a circle in three dimensions. Just as a circle, sphere is also perfectly round and is defined as the locus of points that lie at the same distance say ‘r’ from a fixed point in three dimensional space. This ‘r’ is called as the radius of the sphere and this fixed point is called the center of the sphere. The general equation of sphere in 3D is

(x-a)2 + (y-b)2 + (z-c)2 = r2, where (a, b, c) is the center and r is the radius.

http://cdn1.askiitians.com/Images/201423-134112362-7068-sphere.gif

Illustration: The number of all possible triplets (a1, a2, a3) such that a1 + a2 cos(2x) + a3sin2x = 0 for all x is zero/infinite/one/ three?

Solution: The given condition is a1 + a2 cos(2x) + a3 sin2x = 0

If we put x = 0 and x = π/2 we get

a1 + a2 = 0 and a1 – a2 + a3 = 0

Hence, a2 = – a1 and a3 = a2 – a1 = -2a1

And so this leads us to the equation of the form

 a1 – a1 cos(2x) – 2a1 sin2x = 0 for all x

or a1 (1 – cos(2x) – 2 sin2x) = 0 for all x

which is satisfied for all values of a1

Hence, infinite number of triplets (a1-a1-2a1) are possible.

Illustration: Find the value of k such that (x-4)/1 = (y-2)/1 = (z-k)/2 lies in the plane 2x – 4y + z = 7.

Solution: Given equation of the straight line is  

(x-4)/1 = (y-2)/1 = (z-k)/2

Since the line lies in the plane 2x – 4y + z = 7

Hence, the point (4, 2, k) must satisfy the plane which yields

8-8+k = 7

Hence, k = 7.
askIITians is a platform where students get the opportunity of asking any kind of doubts on topics like shortest distance between two lines, equation of plane parallel to xy plane or direction cosines of line along with the solutions of various 3D coordinate geometry problems.

Download here Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes in PDF Format 

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes

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