Question

find d.c of the line whose equation is

Answer 1

Given equation of line is,

$\longrightarrow\dfrac{x-3}{2}=\dfrac{y-4}{3}=\dfrac{z-1}{5}$

The line in vector form is,

$\longrightarrow\vec r=\left(3\,\hat i+4\,\hat j+\hat k\right)+\lambda\left(2\,\hat i+3\,\hat j+5\,\hat k\right)$

Clearly the vector $2\,\hat i+3\,\hat j+5\,\hat k$ is a direction ratio of the line.

$\longrightarrow\vec{b}=2\,\hat i+3\,\hat j+5\,\hat k$

Its magnitude is,

$\longrightarrow b=\sqrt{2^2+3^2+5^2}$

$\longrightarrow b=\sqrt{38}$

Hence, the direction cosine of the line is,

$\longrightarrow\underline{\underline{\hat b=\pm\dfrac{1}{\sqrt{38}}\left(2\,\hat i+3\,\hat j+5\,\hat k\right)}}$