Formula derivation properties and solved examples of direction cosines
The direction cosines are values of the angles of the three cosines of a vector that are made with the coordinate axes. Equivalently, another way to think of direction cosines is to see them as the components in a correlation of the unit vector signalling in the same direction. They have useful applications for developing direction cosine matrices that represent one set of orthonormal basis vectors in respect of another set, or for representing a known vector on a different basis.
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Direction Ratios of a Line
The fundamental concepts of 3-D geometry contribute to direction cosines and direction ratios. Direction cosines of a line that passes through the origin forming the angles with the coordinate axes are nothing but the numbers that are proportional to the direction cosines.
Direction Cosines in Three-Dimensional Geometry
In 3-D geometry, we have 3 axes: namely, the x, y, and z-axis. Suppose that a line OP passes through the origin in a 3D space. Then, the line will form an angle each with the x, y and z-axis respectively.
The cosines of each of these angles that the line creates along the x-axis, y-axis, and z-axis respectively are known as the direction cosines of the line in three-dimensional geometry. Usually, it is customary to express these direction cosines using the respective letters l, m, and n.
However, you need to remember that these cosines can only be found once we have discovered the angles that the line forms with each of the axes. Also, it is interesting to know that if we reverse the direction of this line, the angles will surely alter.
As a result, the direction cosines i.e. the cosines of these angles will also not be similar once the direction of the line has been reversed. Let us now consider a little different situation where our line does not pass through the point of origin (0, 0, and 0).
Direction Cosines in Case a Line Does Not Pass Through The Origin
You might be thinking over how the direction cosines are to be identified given that the line does not pass through the origin. It is simple. We will have to take into consideration another fictitious (assumed) line parallel to our line in a way that the 2nd line passes through the origin.
Now, the angles that this line forms with the three axes i.e. x-axis, y-axis, z-axis will be similar to that formed by our original line and thus the direction cosines of the angles created by this fictitious line with the axes will be similar for our original line too.
Key Takeaways Direction Cosines
We already know that [l = cos α], [m = cos β] and [n = cos γ] and we are also aware that -1 < cos x < 1 ∀ x ∈ R, so ‘l’, ‘m’ and ‘n’ are real numbers with values fluctuating between -1 to 1. Thus, the value of direction cosine’s ∈ [-1, 1].
The direction cosine of x, y and z axes are [1, 0, 0], [0, 1, 0] and [0, 0, 1] respectively.
The dc’s of a line parallel to any coordinate axis are equivalent to the dc of the corresponding axis.
The three angles formed along the x-axis with the coordinate axis are 0 degrees, 90 degrees and 90 degrees. Thus, the direction cosine turns to cos 0o, cos 90o, and cos 90o i.e. [1, 0, 0].
In case the assigned line is reversed, then the dc will be cos [π – α], cos [π – β], cos [π – γ]. Thus, a line can have two sets of direction cosines as per its direction.
The direction cosines are linked by the relation l2 + m2 + n2 = 1.
The direction cosines of two parallel lines are always similar to one another.
Direction ratios are directly in proportion to direction cosines and thus for a given line, there can be innumerable many direction ratios.
FAQs on Direction Cosines in Three Dimensional Geometry
1. What are direction cosines?
The direction cosines of a line are the cosines of the angles that the line makes with the positive x, y, and z axes in three-dimensional space. If a line makes angles α, β, and γ with the x, y, and z axes respectively, then its direction cosines are l = cosα, m = cosβ, and n = cosγ.
- They describe the orientation of a line in 3D geometry.
- They are always real numbers between −1 and 1.
- They satisfy an important identity: l² + m² + n² = 1.
2. What is the formula for direction cosines?
The formula for direction cosines of a line with direction ratios a, b, c is l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), and n = c/√(a²+b²+c²).
- First find the magnitude: √(a² + b² + c²).
- Divide each component by this magnitude.
- The resulting values l, m, n are the direction cosines.
3. What is the relationship between direction cosines and direction ratios?
The direction ratios are proportional to the direction cosines, and direction cosines are the normalized form of direction ratios.
- If a, b, c are direction ratios, then direction cosines are:
l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²). - Direction ratios need not satisfy any condition.
- Direction cosines always satisfy l² + m² + n² = 1.
4. Why do direction cosines satisfy l² + m² + n² = 1?
Direction cosines satisfy l² + m² + n² = 1 because they are the components of a unit vector.
- Let a line have direction ratios a, b, c.
- Then l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²).
- Squaring and adding gives (a²+b²+c²)/(a²+b²+c²) = 1.
5. How do you find the direction cosines of a line joining two points?
To find direction cosines of a line joining points A(x₁,y₁,z₁) and B(x₂,y₂,z₂), first find direction ratios and then normalize them.
- Direction ratios = (x₂−x₁, y₂−y₁, z₂−z₁).
- Magnitude = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
- Direction cosines = each difference divided by the magnitude.
6. What are the direction cosines of a line parallel to the coordinate axes?
The direction cosines of a line parallel to a coordinate axis have one value ±1 and the other two equal to 0.
- Parallel to x-axis: (±1, 0, 0).
- Parallel to y-axis: (0, ±1, 0).
- Parallel to z-axis: (0, 0, ±1).
7. How do you find the angle between two lines using direction cosines?
The angle θ between two lines is given by cosθ = l₁l₂ + m₁m₂ + n₁n₂, where l, m, n are their direction cosines.
- Multiply corresponding direction cosines.
- Add the products.
- Take inverse cosine to find θ.
8. Can direction cosines be negative?
Yes, direction cosines can be negative because they are cosines of angles measured with respect to the positive coordinate axes.
- If the angle with an axis is greater than 90°, its cosine is negative.
- Example: cos120° = −1/2.
- However, they must still satisfy l² + m² + n² = 1.
9. How do you check if given numbers are direction cosines?
To check if numbers l, m, n are direction cosines, verify that l² + m² + n² = 1.
- Square each number.
- Add the squares.
- If the sum equals 1, they are valid direction cosines.
10. What is an example of finding direction cosines?
An example of finding direction cosines is for a line with direction ratios 2, 3, 6.
- Magnitude = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
- l = 2/7, m = 3/7, n = 6/7.
- Check: (2/7)² + (3/7)² + (6/7)² = 4/49 + 9/49 + 36/49 = 1.
