Question

The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the three concurrent edges of the cube are coordinate axes.)
a.\[\dfrac{2}{{\sqrt 3 }},\dfrac{2}{3},\dfrac{2}{3}\]
b.\[1,1,1\]
c.\[2, - 2,1\]
d.\[1,2,3\]

Answer 1

Hint: Concurrent edges are the set of lines or curves if they all intersect each other at the same point. The point at which all the edges intersect is called the point of concurrency. In the given question the diagonals of the cube join at the origin where origin represents the point of concurrency. Construct a diagram of the cube and let the side of the cube be of \[a\] units. Further plot the points on each axis and also suppose the random point which is \[a\] units away from the origin and lies in all planes. Thus, find the direction ratio between the point which lies in all the planes and the origin.

Complete step-by-step answer:
Consider the figure below,

From the figure which represents the cube, we can notice that we have let the side of the cube be \[a\].
Next, we will find the points at \[x\]-axis, \[y\]-axis, \[z\]-axis and one point which lies in all the planes.
So, the point which lies in all the planes is \[A = \left( {a,a,a} \right)\].
Next, the point which lies on the \[x\]-axis is \[B = \left( {a,0,0} \right)\]
Next, the point which lies on the \[y\]-axis is \[C = \left( {0,a,0} \right)\]
Further, the point which lies on the \[z\]-axis is \[D = \left( {0,0,a} \right)\]
In the question, we are given that the 3 concurrent edges of the cube are coordinate axis which means that \[x\]-axis, \[y\]-axis, \[z\]-axis represents the concurrent edges which have the common intersection point which is origin as all the three axis are getting intersected at origin only.
Thus, the point \[O = \left( {0,0,0} \right)\] represents the concurrency point and the three axes are the concurrent edges.
The opposite corner which is joining the origin is represented by point \[A = \left( {a,a,a} \right)\]
Thus, we need to find the direction ratio of the diagonals of the cube joining the origin to the opposite corner.
Direction ratios represent the ratio of each coordinate to the total magnitude
Now, we will find the direction ratio of line \[OA\].
Thus, the direction ratios of diagonal \[OA\] are \[\left( {a - 0,a - 0,a - 0} \right)\]
This gives us, \[\left( {a,a,a} \right)\]
Therefore, \[\left( {1,1,1} \right)\]
Hence the direction ratios of the diagonal of the cube joining the origin to the opposite corner are \[\left( {1,1,1} \right)\].
The option is incorrect as the ratio is \[\left( {1,1,1} \right)\].

Note: Construct a cube and represent the 3 axes with the points marked on it and the common point which lies in all the 3 planes. The Concurrent edges are the set of lines or curves if they all intersect each other at the same point. The three axes represent the concurrent edges in the figure and at the origin all the 3 edges get intersected.