Direction Ratios, Direction Cosine & Direction Angles in Vector

Direction ratios, direction cosines, and direction angles are used to describe the orientation of a line or vector in three-dimensional geometry. These ideas help students connect coordinates, vectors, and angles, and they are useful in later topics such as the equation of a line and the angle between two lines.

The angles made by a vector with the positive directions of the X-axis, Y-axis and Z-axis are called direction angles of the vector, denoted by α, β, and γ.

\[ l = \cos\alpha, \quad m = \cos\beta, \quad n = \cos\gamma, \]

\[ \boxed{\,\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1\,} \]

If α, β and γ are the direction angles of a vector, then the cosines of these angles, i.e.

l = cos⁡α, m = cos⁡β, n = cos⁡γ 

are called the direction cosines of the vector.

If \[ P(x,y,z) \] is a point and \[ \vec{OP} \] is its position vector, then

\[ r = |\vec{OP}| = \sqrt{x^{2} + y^{2} + z^{2}} \]

and

\[ \boxed{\,\cos\alpha = \frac{x}{r}, \qquad \cos\beta = \frac{y}{r}, \qquad \cos\gamma = \frac{z}{r}\,} \]

If l, m, n are direction cosines of a line and if a, b, c are real numbers such that \[\frac{\mathrm{a}}{l}=\frac{\mathrm{b}}{\mathrm{m}}=\frac{\mathrm{c}}{\mathrm{n}}=\lambda,\] then a, b, c are called direction ratios of that line.

\[ \boxed{\,\frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}}, \qquad \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}}, \qquad \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}}\,} \]

• Direction angles are the angles a line makes with the positive coordinate axes.

• Direction cosines are \[\cos \alpha\], \[\cos \beta\], and \[\cos \gamma\].

• If direction cosines are (l, m, n), then \[l^2 + m^2 + n^2 = 1\].

• Direction ratios are any numbers proportional to direction cosines.

• If direction ratios are (a, b, c), then corresponding direction cosines are:

• For points \[A(x_1, y_1, z_1)\], \[B(x_2, y_2, z_2)\], direction ratios of AB are \[(x_2 - x_1, y_2 - y_1, z_2 - z_1)\].

• Angle between two lines can be found using either direction cosines or direction ratios.