Question

If OA is equally inclined to OX, OY, and OZ and if A is \[\sqrt 3 \] units from the origin, then A is

A. \[\left( {3,3,3} \right)\]
B. \[\left( { - 1,1, - 1} \right)\]
C. \[\left( { - 1,1,1} \right)\]
D. \[\left( {1,1,1} \right)\]

Answer 1

Hint:In this question, we need to find the coordinates of A. For this, we have to assume that the coordinates of A are \[\left( {a,a,a} \right)\]. After that we will use the given condition such as the distance of A from the origin is \[\sqrt 3 \] units using the distance formula to get the desired result.

Formula used:
The distance between two points A and B whose coordinates are \[\left( {{x_1},{y_1},{z_1}} \right)\]and \[\left( {{x_2},{y_2},{z_2}} \right)\]is given by
\[d\left( {A,B} \right) = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} \]
Complete step-by-step answer:
We know that OA is equally inclined to OX, OY and OZ.
Let the coordinates of A be \[\left( {a,a,a} \right)\].
We have been given that \[OA = \sqrt 3 \]
Here, the coordinates of the origin O is \[\left( {0,0,0} \right)\]
Also, the coordinates of A is \[\left( {a,a,a} \right)\]
Here, we can say that \[\left( {0,0,0} \right) \equiv \left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {a,a,a} \right) \equiv \left( {{x_2},{y_2},{z_2}} \right)\]
So, by using distance formula, we get
\[d\left( {O,A} \right) = \sqrt {{{\left( {a - 0} \right)}^2} + {{\left( {a - 0} \right)}^2} + {{\left( {a - 0} \right)}^2}} \]
But \[d\left( {O,A} \right) = \sqrt 3 \]
Thus, we get
\[\sqrt 3 = \sqrt {{{\left( a \right)}^2} + {{\left( a \right)}^2} + {{\left( a \right)}^2}} \]
\[\sqrt 3 = \sqrt {3{{\left( a \right)}^2}} \]
\[\sqrt 3 = \sqrt 3 \times \sqrt {{{\left( a \right)}^2}} \]
\[a = \pm 1\]
That means there are two possibilities here.
The coordinates of A are \[\left( {1,1,1} \right)\]or \[\left( { - 1, - 1, - 1} \right)\].


Therefore, the correct option is (D).

Additional Information: We can say that the distance formula is defined as the algebraic expression that provides the distances between pairs of points in terms of their coordinates. We can find the distance formula for finding the distance between two points in the XYZ space and XY plane also. That is 3D space and 2D space. The 3d distance formula is an extended part of the 2d distance formula. 

Note: Many students make mistakes in applying the distance formula specifically, in the calculation part. They may get confused with signs and taking the right coordinates in right place.