##### Chapter 4

## Vectors And Three Dimensional Geometry

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

#### Introduction

**Vectors And Three Dimensional Geometry**while solving your maths problems in your geometry class. But have you faced difficulty in solving them easily? If yes, then we are here to aid you in understanding the core concept behind them. In this article, we will discuss everything about

**Vectors And Three Dimensional Geometry.**We hope that this article will help clear your doubts. This topic is very important to understand different types of shapes and figures in geometry.

It depends on a two-dimensional plane consisting of the x-axis and the y-axis. Opposite to one another, the axes partition the plane into four areas. Each part is known as a quadrant. There are four quadrants. Namely:-

**Quadrant I**

**Quadrant II**

**Quadrant III**

**Quadrant IV**

The two-dimensional plane where the

- the x-axis is the horizontal axis
- the y-axis is the vertical axis

A point in the plane is characterized as an arranged pair (x,y), with the end goal that x is controlled by its flat separation from the beginning, and y is dictated by its vertical separation from the source.

We can find** Distance Between 2 Points **by using the distance formula. The formula is **d=√((x_2-x_1)²+(y_2-y_1)²). **Here, the Pythagorean theorem is applied. The best way to learn this formula is to set up a correct triangle and utilize the Pythagorean hypothesis at whatever point they need to discover the distance between two focuses.

As an uncommon instance of the distance equation, assume we need to know the distance of a point (x,y) to the origin. As per the distance recipe, this is √(x−0)2+(y−0)2=√x2+y2. A point (x,y) is a way off r from the origin if and just if √x2+y2=r, or if we square the two sides: x2+y2=r2.

In Maths, a triangle is a three-sided polygon that has three edges and three vertices. The territory of the triangle is a proportion of the space canvassed by the triangle in the two-dimensional plane. At the point when we have vertices of the triangle, and we need to discover the zone of the triangle, we can follow the following steps:-

- The plot focuses on a chart.
- Take the vertices in a counterclockwise direction. In any case, the equation gives a negative worth.
- Use the equation A = Hbb/2.

The cosines of a line are the cosines of the points which the line makes with the positive bearings of the facilitated coordinate axes.

Here it is noted that the direction cosines for any line should be exceptional. Be that as it may, there are limitlessly numerous arrangements of bearing proportions since numbers proportions are only a set of any three numbers relative to the direction cosines.

**Parallel** **lines** will be lines in a plane that are consistently a similar distance separated. **Parallel **lines won’t ever meet. Parallel planes will be planes in the very three-dimensional space that won’t ever meet. Parallel lines can be horizontal, vertical, or diagonal.

When two straight lines are plotted on the coordinate plane, we can tell if they are parallel from the slope of each line. If the slopes are the same, then the lines are parallel.

The projection here signifies “The portrayal of a figure or strong on a plane as it would look from a specific course”. The **projection of line** can be of two types:-

**Not belonging to a line:**The**projection of line**when it is outer to the line.**Belonging to a line:**The**projection of line**when it is on the line.

**Question 1. What is a three-dimensional vector?**

**Answer. **A three-dimensional vector is a line segment in three-dimensional space running from point A to point B. Each vector has a magnitude and direction.

**Question 2. What is a Vector?**

**Answer. ** A vector is a coordinated line section whose length is the extent of the vector and with a bolt showing the heading. The course of the vector is from its tail to its head.

**Question 3. What are the examples of parallel lines in real life?**

**Answer. **We can see parallel lines in real life in a notebook, railway tracks, zebra crossing, etc.** **

**Question 4. What are the main properties of parallel lines?**

**Answer. **The following are the main properties of parallel lines:-

- The comparing points are equivalent.
- The vertically inverse points are equivalent.
- The other inside points are equivalent.
- The other outside points are equivalent.
- The pair of inside points on a similar side of the cross-over is strengthening.

**Question 5. Define the Cartesian vector form.**

**Answer. **The vector, being the amount of the vectors and, is subsequently. This equation, which communicates j, k, x, y, and z, is known as the Cartesian portrayal of the vector in three-dimension**.**

Many questions are asked from **Vectors And Three Dimensional Geometry **in class 12 board exams. Formulas are critical so understand them by practising as many numerical as possible. Topics are easy and can be completed effortlessly. For your professional growth, it is important to have a proper understanding of this topic. You can check the MSVgo app to know more about the topic. The MSVgo philosophy is to enable a core understanding of any concept. MSVgo is a video library that explains concepts with examples or explanatory visualizations or animations. Check out videos on MSVgo to understand the concept, behind **Vectors And Three Dimensional Geometry.**