Direction Cosines and Direction Ratios of a Line | Definitions & Formulas
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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.
What are Direction Cosines and Direction Ratios?
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.
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Direction Cosines: The direction cosine is the cosine of the angle subtended by this line with the x-axis, y-axis, and z-axis respectively. If the angles subtended by the line with the three axes are α, β, and γ, then the direction cosines are cos(α), cos(β), and cos(γ) respectively.
The direction cosines for a vector
The direction cosines are also represented by
Direction Ratios: Direction ratios are the components of a vector along the x-axis, y-axis, and z-axis respectively.
The direction ratios of a vector
Direction ratios of a vector line joining two points
Direction Cosines of the Line Passing Through Two Points
Let
Let l,m, and n are the direction cosines of the line PQ, and let it makes angles α, β, and γ with x-axis, y-axis, and z-axis respectively.
Draw a perpendicular from
Let these perpendiculars meet the
Draw a perpendicular from
Now in right angle triangle
Therefore,
Similarly, we get
Hence the direction cosines of the line segment joining the points
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Direction Cosines when Line does not Pass Through Origin
When the line does not pass through the origin then how one can find the direction cosines of the line. The answer is simple. Consider another fictitious line parallel to our line such that the second line passes through the origin. Now, the angles made by this line with the three axes will be the same as that made by our original line and hence the direction cosines of the angles made by this fictitious line with the axes will be the same for our original line as well.
Here, the line under question is labeled as OP. It passes through the origin and we are to find out the direction cosines of the line.
Let us assume that the magnitude of the vector is ‘r’ and the vector makes angles α, β, and γ with the coordinate axes. Now, using Pythagoras theorem, we can express the coordinates of the point P(x,y,z) as
where,
Now, we can replace
In the orthogonal system, we can represent
Substitute the values of
By the above statement, it can be said that the direction cosines are the coefficients of the unit vectors
Relationship Between Direction Cosines
Let
Let
Then
From P drawPA,PB,PC perpendicular on the coordinate axis, so that OA=x, OB=y, OC=z.
Also,
From triangle
Similarly,
Hence from equation 1, we get
Hence we have the relationship between the direction cosines as
Difference Between Direction Ratios and Direction Cosines
The difference between direction ratios and direction cosines are listed below:
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Direction Cosines |
Direction Ratios |
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The direction cosine is the cosine of the angle subtended by this line with the |
The direction ratios are the components of a vector along the |
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The direction cosines for a vector |
The direction ratios of a vector |
Direction Cosines and Direction Ratios Formulas
The formulas for direction cosines and direction ratios are tabulated below:
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Description |
Formula |
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Direction ratios of a vector |
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Direction cosines for a vector |
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Direction ratios of a vector line joining two points |
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Direction cosines for a line passing through two points |
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Relationship between the direction cosines |
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Solved Examples of Direction Cosines and Direction Ratios
Example 1: Find the direction cosines and direction ratios of the following vector:
Solution: The direction ratios of
The direction cosinesare
We know
Therefore, the direction cosines are
Example 2: Find the direction cosines of a vector whose direction ratios are
Solution: The direction cosines are
Here
We know
Therefore, the direction cosines are
Example 3: Find the direction cosines of the line joining
Solution: The formula for the direction cosines for a line joining two points is as follows:
DirectionCosines=
Here
Substitute these values in the formula of direction cosines for a line joining two points, we get
DirectionCosines=
Therefore, the direction cosines of the line joining
Example 4: Find the directional cosines for a line that makes
Solution: The directional cosines for the given line are,
Therefore, the directional cosines for the given line are
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FAQs For Direction Cosines and Direction Ratios
What is the difference between direction cosines and direction ratios?
The direction ratio represents the components of a vector or a line with respect to the three-axis, and the direction cosines represent the angle subtended by the line with respect to the three-axis.
How do you find the direction ratio with direction cosines?
The direction ratios of a vector
What is the direction ratio of a vector?
The direction ratios of a vector
What is the meaning of direction cosines?
The direction cosine is the cosine of the angle subtended by this line with the latex]x[/latex]-axis,
Can direction cosines be zero?
Yes, the direction cosines can be zero. The direction cosine of
Can a line have more than one set of direction ratios?
Yes, infinitely many sets of direction ratios exist for a line, all proportional to one another.
Where are direction cosines and ratios used?
They are used in 3D geometry, physics (especially in vectors and motion), engineering, and computer graphics for determining orientation and angles.
Are direction ratios unique for a line?
No, they are not unique. Any scalar multiple of a set of direction ratios also represents the same line.
Can direction cosines be negative?
Yes, direction cosines can be negative depending on the direction of the line. For example, if the line is pointing in the negative x-direction, then the cosine of the angle it makes with the x-axis will be negative.
What is the geometric meaning of direction ratios?
Direction ratios represent a vector along the line. They give the relative amount the line moves along the x, y, and z directions. It’s like saying “for every step in x, take this much in y and z.”