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. The direction ratios for a vector \(\vec A = a\hat i + b\hat j + c\hat k\) is a, b, c 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 direction ratios of a vector \(\vec A = a\hat i + b\hat j + c\hat k\) is (a, b, c) respectively, and these values represent the component values of the vector along the x-axis, y-axis, and 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.

Vector: \(\vec A = a\hat i + b\hat j + c\hat k\)

Direction Ratios: a, b, c

Direction ratios of a vector line joining two points \((x_1, y_1, z_1)\), and \((x_2, y_2, z_2)\) are \((x_2 - x_1, y_2 - y_1, z_2 - z_1)\). 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. 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 \(\overrightarrow A = a \hat i + b \hat j + c \ \hat k\) having direction ratios a, b, c 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 = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\).

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

The direction ratios is used to compare two vectors. The direction ratios of two parallel vectors are in proportion. For two vectors \(\vec A = a_1\hat i + b_1\hat j + c_1\hat k\), \(\vec B = a_2\hat i + b_2\hat j + c_2\hat k\), the direction ratios are \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} =\dfrac{c_1}{c_2}\).

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. For two vectors \(\vec A = a_1\hat i + b_1\hat j + c_1\hat k\), \(\vec B = a_2\hat i + b_2\hat j + c_2\hat k\), the dot product of the vectors is \(\vec A.\vec B = a_1.a_2 +b_1.b_2+c_1.c_2\).

The direction ratios is helpful to find the angle between two vectors. The cos of the angle between two vectors can be computed easily by taking the dot product of the two vectors and dividing it with the product of the magnitude of the two vectors. For two vectors \(\vec A = a_1\hat i + b_1\hat j + c_1\hat k\), \(\vec B = a_2\hat i + b_2\hat j + c_2\hat k\), the angle between the two vectors is Cosθ=\(\dfrac{ a_1.a_2 +b_1.b_2+c_1.c_2}{\sqrt{a_1^2 + b_1^2+c_1^2}.\sqrt{a_2^2+b_2^2+c_2^2}}\).

☛Related Topics

• Vectors
• Angle Between Two Vectors
• Product of Vectors
• Types of Vectors
• Addition Of Vectors

What Is Direction Ratio?

Direction ratios represent the components of a vector, or the components of a line with respect to the x-axis, y-axis, and z-axis respectively. The direction ratio of a vector line \(\vec A = a\hat i + b\hat j + c\hat k\) are a, b, c respectively. The direction ratios are normally represented for a vector or a line in a three-dimensional system.

How To Find Direction Ratios?

The direction ratios can be easily found by observing the given vector. The coefficients of the unit vectors \(hat i\), \(\hat j\), \(\hat k\) in a vector representation are the direction ratios of the vector. For a vector \(\vec A = 5\hat i + 3\hat j + 2\hat k\) the direction ratios are 5, 3, 2.

Which Topic Of Maths Has The Concept Of Direction Ratio?

The topics of three-dimensional geometry and vectors make use of direction ratios. The direction ratio of a point represented in a three-dimensional space is (a, b, c), and the direction ratios of a vector \(\vec A = a\hat i + b\hat j + c\hat k\) is a, b, c respectively.

What Is the Difference Between Direction Ratio And Direction Cosines?

The direction ratio represented 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. The direction ratio of a vector \(\vec A = a\hat i + b\hat j + c\hat k\) is a, b, c, and the direction cosine are Cosα = \(\frac{a}{\sqrt {a^2 + b^2 + c^2}}\), Cosβ = \(\frac{b}{\sqrt {a^2 + b^2 + c^2}}\), Cosγ = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\).

What Is The Use Of Direction Ratios?

The direction ratios are useful to find the relationship between two lines or vectors. The direction cosines of a line or the angle between two lines can be found using the direction ratios. The direction ratio is helpful to find the dot product between two vectors.