Direction Cosine

Direction cosine is the cosine of the angle made by the line in the three-dimensional space, with the x-axis, y-axis, z-axis respectively. Direction cosines can be calculated for a vector or a straight line in a three-dimensional space. It is the cosines of the angle made by the line with the three axes.

Let us learn more about the direction cosine, the relationship between the direction cosines, and the direction cosine of a line connecting two points in a three-dimensional space.

Direction Cosine gives the relation of a vector or a line in a three-dimensional space, with each of the three axes. 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 Cosα, Cosβ, Cosγ respectively.

The direction cosines for a vector \(\overrightarrow A = a \hat i + b \hat j + c \ \hat k\) is Cosα = \(\frac{a}{\sqrt {a^2 + b^2 + c^2}}\), Cosβ = \(\frac{b}{\sqrt {a^2 + b^2 + c^2}}\), Cosγ = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\). The direction cosines are also represented by l, m, n and we often represent the direction cosines as l = \(\frac{a}{\sqrt {a^2 + b^2 + c^2}}\), m = \(\frac{b}{\sqrt {a^2 + b^2 + c^2}}\), n = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\).

The direction cosine for a point A (a, b, c) in a three-dimensional space is the direction cosine of the line connecting this point with the origin O. The direction cosine of the line OA is l = \(\frac{a}{\sqrt {a^2 + b^2 + c^2}}\), m = \(\frac{b}{\sqrt {a^2 + b^2 + c^2}}\), n = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\).

The direction cosines of a line joining the point(a, b, c) with the origin is Cosα, Cosβ, Cosγ respectively, and the distance of this point from the origin be r. Here the values of these direction cosines areCosα = a/r, Cosβ=b/r, Cosγ=c/r.The value of distance r is \(\sqrt {a^2 + b^2 + c^2}\). Here we aim at finding the relationship between the direction cosines of this point. Let us square and add the direction cosines of the point.

Cos²α + Cos²β + Cos²γ = a²/r² + b²/r² + c²/r²

Cos²α + Cos²β + Cos²γ = (a² + b² + c²)/r²

But we have r² = \(a^2 + b^2 + c^2\). Substituting this in the above expression we have.

Cos²α + Cos²β + Cos²γ = r²/r²

Cos²α + Cos²β + Cos²γ = 1

Let us now consider l = Cosα, m = Cosβ, n = Cosγ. Hence we have the relationship between the direction cosines as l² + m² + n² = 1.

The direction cosine of a line joining two points (\(x_1, y_1, z_1\)), and (\(x_2, y_2, z_2\)), can be easily computed using the direction ratios of the lines joining these two points, and by finding the distance between these two points. The direction ratio of the line joining these two points is \(x_2 - x_1\), \(y_2 - y_1\), \(z_2 - z_1\). And the distance between these two points is \(\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2} \).

The direction cosine of a line is calculated by dividing the respective direction ratios with the distance between the two points. The formula for the direction cosines for a line joining two points is as follows.

Direction Cosines = \(\left(\dfrac{x_2 - x_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{y_2 - y_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{z_2 - z_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}\right)\)

Related Topics

The following topics help in a better understanding of direction cosines.

• Direction of a Vector
• Unit Vector
• Position Vector
• Types of Vectors
• Components of a Vector

How Do You Find Direction Cosines?

The direction cosine for a vector or a line is a three dimension space can be calculated from the direction ratio, and the magnitude of the vector. The direction cosine for a point A (a, b, c) in a three-dimensional space is the direction cosine of the line connecting this point with the origin O. The direction cosine of the line OA is l = \(\frac{a}{\sqrt {a^2 + b^2 + c^2}}\), m = \(\frac{b}{\sqrt {a^2 + b^2 + c^2}}\), n = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\).

What Is Direction Cosine and Direction Ratios?

The direction cosines can be calculated by dividing the respective direction ratios with the magnitude of the vector. The direction ratios of a line joining the point point A (a, b, c) with the origin in the three dimensional frame is a, b, c respectively, and the direction ratios are \(\left(\frac{a}{\sqrt {a^2 + b^2 + c^2}}, \frac{b}{\sqrt {a^2 + b^2 + c^2}}, \frac{c}{\sqrt {a^2 + b^2 + c^2}} \right)\).

What Are the Direction Cosines of 2i + 3j + 4k?

The direction ratios of the vector 2i = 3j + 4k is 2, 3, 4, and the magnitude of the vector is \(\sqrt{2^2 + 3^2 + 4^2} = \sqrt{29}\). Hence the direction cosines of the vector is (2/√29, 3/√29, 4/√29).