How to find direction ratios of a line

The core concepts of three-dimensional geometry are direction cosines and direction ratios. What are direction cosines of a line that passes through the origin that makes angles with the coordinate axes? This lesson helps you understand the concepts of direction cosines and direction ratios which are nothing but numbers proportional to the direction cosines. A solved problem, in the end, will help you understand the concepts better.

Read about the direction cosines and direction ratio of a line. Learn the formula of the direction cosines and direction ratio of the line. The number that is proportional to the direction of cosine is called the direction ratio of the line. The direction ratio of the line is represented by a,b,c. Consider the point P x, y, z having coordinates 1, 2, 3.

How to find direction ratios of a line

Direction ratio helps in knowing the components of a line or a vector with reference to the three-axis, the x-axis, y-axis, and z-axis respectively. The direction ratios helps in finding the direction cosines, the angle between two lines, or the dot product of two vector. Let us learn more about the direction ratios, their relationship with direction cosines, and the uses of direction ratios, with the help of examples, FAQs. Direction ratios are the components of a vector along the x-axis, y-axis, z-axis respectively. The number of direction ratio depends on the dimension of the space. For a line in a two-dimensional space, there are two direction ratios, and for a line in a three-dimensional space, there are three direction ratios. The direction ratios are useful to find the direction cosines of a line. There can be an infinite set of direction ratios for a given line, and the direction ratios of two parallel lines are in proportion. The direction ratios help in finding the direction cosines of a line. The direction cosine is the cosine of the angle subtended by this line with the x-axis, y-axis, and z-axis respectively. The direction ratios is used to compare two vectors. The direction ratios of two parallel vectors are in proportion. The direction ratio is useful to find the dot product of two vectors. The dot product of two vectors is the summation of the product of the respective direction ratios of the two vectors.

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Direction cosine is the cosine of the angle made by the line in the three-dimensional space, with the x-axis, y-axis, and z-axis and Direction ratio helps in knowing the components of a line or a vector with reference to the three-axis, the x-axis, y-axis, and z-axis. Direction cosines and direction ratios come into existence as soon as a vector comes into existence in a three-dimensional coordinate space. Direction ratios are the components of a vector along the x-axis, y-axis, and z-axis, and direction cosine is the cosine of the angle subtended by this line with the x-axis, y-axis, and z-axis. The direction ratios are useful to find the direction cosines of a line. When the line does not pass through the origin then how one can find the direction cosines of the line.

Before discussing directional cosines of a vector, let us discuss the position vector. As the name suggests, a position vector indicates the position of any point relative to any reference origin. Consider any arbitrary point in three-dimensional space having the co-ordinates x,y,z as with respect to the origin O 0,0,0. The following figure represents a vector P in space with O as reference origin. These angles are known as direction angles and on taking the cosine of these angles we get direction cosines. Taking direction cosines makes it easy to represent the direction of a vector in terms of angles with respect to the reference. This is the mathematical representation of cosines of the directions of a vector. The above equations can be rewritten as,. The co-ordinates of the point P may also be expressed as the product of the magnitude of the given vector and the cosines of direction on the three axes, i. We can clearly see that lr,mr,nr are in proportion to the direction cosines and these are called as the direction ratios and they are denoted by a,b,c.

How to find direction ratios of a line

Usually, for three-dimensional geometry, we rely on the three-dimensional Cartesian plane. Vectors can also be used to describe the lines and the angles they make with the axis. How should we describe a line passing through origin making an angle with different axes? We define them using cosine ratios of the line. While working with three-dimensional geometry used in so many applications such as game designing , it is required to describe the significance of the line present in 3-D space. The figure given below shows the line OA passing through the origin and all the angles it makes with the coordinate axes. Note: The line can extend in both the directions, and there can be two sets of angles that it makes with coordinate axes. Now the signs of the directional cosines will be reversed. So, to have unique directional cosines, we choose to make it a directed line.

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Report An Error. Enhance the article with your expertise. We draw a perpendicular from D to the x-axis. Help us improve. Important Links. What is the difference between the functions direction cosine and direction ratios? If it is also very simple, we can create the friction line as the 2nd line passes through the origin instead of the original line. Kindergarten Worksheets. Suggest changes. Also, it is interesting to note that if we reverse the direction of this line, the angles will obviously change. Similar Reads. Direction Ratio Direction ratio helps in knowing the components of a line or a vector with reference to the three-axis, the x-axis, y-axis, and z-axis respectively. Direction ratio helps in knowing the components of a line or a vector with reference to the three-axis, the x-axis, y-axis, and z-axis respectively. Do parallel lines have the same direction ratios? Substitute these values in the formula of direction cosines for a line joining two points, we get.

The direction cosines of a directed line are the cosines of the angles which the line makes with the positive directions of the coordinate axes. Consider a line OL as shown, passing through the origin O.

Please Login to comment Previous Section Formula - Vector Algebra. The direction line gets reversed if we reverse the direction of the line OQ. We will be shedding light on all these significant topics over here. Article Tags :. What is the relation between three angles that the line makes with origin? While working with three-dimensional geometry used in so many applications such as game designing , it is required to describe the significance of the line present in 3-D space. Read about the direction cosines and direction ratio of a line. The direction cosines of a line or the angle between two lines can be found using the direction ratios. But hurry up, because the offer is ending on 29th Feb! The directional cosines of the lines are assumed to be l, m, and n. You will be notified via email once the article is available for improvement.