Question

The coordinates of the point P are \[(x,y,z)\] and the direction cosine of the line OP when O is the origin, are l, m, n. If \[OP = r\] then find the relation between \[l,m,n\] and \[x,y,z\] .
A.\[l = x,m = y,n = z\]
B. \[l = xr,m = yr,n = zr\]
C. \[x = lr,y = mr,z = nr\]
D. None of these

Answer 1

Hints We know the formula of direction cosine of a line, apply the formula and obtain the distance from O to P to obtain the value of r. Then establish a relation between the coordinate of the given point and the given direction cosines.

Formula used
The formula of direction cosine of the line OA, where O is the origin and \[A(a,b,c)\] is any point,
then,
\[\cos \alpha = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]
\[\cos \beta = \dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]
\[\cos \gamma = \dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]
And the distance formula of OA is,
\[OA = \sqrt {{{\left( {a - 0} \right)}^2} + {{\left( {b - 0} \right)}^2} + {{\left( {c - 0} \right)}^2}} \]
=\[\sqrt {{a^2} + {b^2} + {c^2}} \]

Complete step by step solution
Given that the direction cosines are \[l,m,n\] and the given point is \[P(x,y,z)\].
Therefore,
\[l = \dfrac{x}{{\sqrt {{x^2} + {y^2} + {z^2}} }}\] ---(1)
\[m = \dfrac{y}{{\sqrt {{x^2} + {y^2} + {z^2}} }}\]-----(2)
\[n = \dfrac{z}{{\sqrt {{x^2} + {y^2} + {z^2}} }}\]-----(3)
Now, the distance from O to P is,
\[OP = \sqrt {{{\left( {0 - x} \right)}^2} + {{\left( {0 - y} \right)}^2} + {{\left( {0 - z} \right)}^2}} \]
\[ = \sqrt {{x^2} + {y^2} + {z^2}} \]
But, given that \[OP = r\], hence,
\[r = \sqrt {{x^2} + {y^2} + {z^2}} \]
Therefore, from the equation (1), (2) and (3) we have,
\[l = \dfrac{x}{r}\]
That is, \[x = lr\]
\[m = \dfrac{y}{r}\]
That is, \[y = mr\]
\[n = \dfrac{z}{r}\]
That is, \[z = nr\]
Hence, the required relation is \[x = lr,y = mr,z = nr\].
The correct option is C.

Note Students sometimes skip the part in which we are showing the distance between the point P and the origin and after that, we have done the substitution, which is not the correct way to solve, it is obvious that \[r = \sqrt {{x^2} + {y^2} + {z^2}} \], but we need to show this in our calculation.