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Maths > Introduction to Three Dimensional Geometry > Introduction to 3D Geometry
Introduction to Three Dimensional Geometry

Introduction to 3D Geometry

One of the most powerful tools in any engineer or scientist’s toolkit is 3D geometry. 3D geometry comes to picture into model real-world quantities such as velocity, fluid flows, electrical signals, and many other physical quantities. In this article, we will first attempt at understanding a 3D system.  Later we will move on to a basic understanding of how to perform various mathematical operations is a 3D space.

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Understanding the 3D Geometry

We can understand any 3D space in terms of 3 coordinates- x, y, and z.

3d geometry

Vectors

Vectors are the Fundamental Unit of 3D Operations. In physics, Vectors are quantities having both magnitude and distance. In mathematics, especially in 3D geometry, a vector is a directed entity that connects 2 or more points. A position vector is a special type of vector that connects the origin O(0,0,0) to the point, as shown below:

3d geometry

Here, we have the position vector P, defined by an arrow from O(0,0,0) to P(1,1,1). Note that the starting point of the vector is defined as the tail of the vector, and the ending point of the vector is defined as the head of the vector. The direction of any vector is always from tail to head.

In physics, we have a special kind of vector known as a displacement vector. A displacement vector is used to quantify the vector between 2 points. A position vector is actually a special kind of displacement vector.

Magnitude of a Vector

The magnitude of a vector is a measure of how long a vector is. Consider a vector whose tail is given by T(x1, y1, z1) and whose head is given by H(x2, y2, z2). If we define this vector as V, then the magnitude of the vector is denoted as |V|, where:

|V| = √{(x2-x1)2+(y2-y1)2+(z2-z1)2}

Example: Calculate the magnitude of a vector whose tail is given by (1,2,3), and whose head is given by (4,5,6)

Solution: In the given example, we have:

x1 = 1, x2 = 4, y1 = 2, y2 = 5, z1 = 3, z2 = 6

|V| =√{(x2-x1)2+(y2-y1)2+(z2-z12}

|V| = 3√3 units

Example: Calculate the magnitude of a vector whose tail is  (-1,-4,3), and whose head is  (4,7,-2)

Solution: In the given example, we have:

x1 = -1, x2 = 4, y1 = -4, y2 = 7, z1 = 3, z2 = -2

|V| = √{(x2-x1)2+(y2-y1)2+(z2-z1)2}

So, |V| = √((4-(-1))2+(7-(-4))2+((-2)-3)2}

|V| = √171 = 3√19 units

A unit vector is any vector that has a magnitude of 1.

Components of a Vector

If we consider a Cartesian coordinate system, any vector can be defined in terms of 3 components. We can define the notation of any vector as:

v = |x| \( \hat{i} \) + |y| \( \hat{j} \) + |z| \( \hat{k} \)

Where \( \hat{i} \),  \( \hat{j} \), and \( \hat{k} \) are the unit vectors along the x, y and z-axes respectively, and |x|, |y|, and |z| denote the length of the components of the vector along these axes respectively. The magnitude of a vector V, is |x| \( \hat{i} \) + |y| \( \hat{j} \) + |z| \( \hat{k} \) can be denoted as |V|, where:

|V| = √(|x|2+|y|2+|z|2)

Example: Calculate the x, y, and z components of the vector u = 3\( \hat{i} \) + 4\( \hat{j} \) + 5\( \hat{k} \) as well as the magnitude of the vector.

Solution: In the above problem, we have |x| = 3, |y| = 4, and |z| = 5

x-component = 3, y-component = 4, z-component = 5

|u| = √(32 + 42 + 52) = 5√2

More Solved Examples for You

Question 1: The point (0, -2, 5) lies on the:

  1. X-axis
  2. Y-axis
  3. XY-plane
  4. YZ-plane

Answer : Option D. Given Point is (0, -2, 5). The X co-ordinate of the point is 0. Thus, the point lies in the YZ-plane.

Question 2: What are three-dimensional shapes?

Answer: Three-dimensional (3D) shapes refer o all those shapes that are composed of a 3D geometric shape. Furthermore, these shapes occupy space and have a volume (means they can be filled with liquid). Some common examples of 3D shapes are prism, sphere, cone, cylinder, cube, rectangle, etc.

Question 3: What are the first three dimensions?

Answer: These are space dimensions such as x-y-z, hence they are width, length, and height. The zero dimensions are that dimension in which there is the only point, the first dimension has only length, the second dimension has length and breadth and so on.

Question 4: What is a 3D coordinate system?

Answer: It refers to a Cartesian coordinate system, which is formed by a point called the origin. Moreover, it basically consists of three mutually perpendicular vectors. And these vectors define the three coordinate axes: the x-, y- and z-axis. Besides, we also know them as the abscissa, ordinate and applicate axis, respectively.

Question 5: Is time the 4th dimension?

Answer: Yes, we regard time as the fourth dimension, although in a distinctive sense than only going from 2D to3D. We lookto time as the 4th dimension because of relativity, as we have to incorporate the time synchronize in defining events fully in space along with the time called the space-time coordinate.

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