(B) Let the side length of the cube be $a$.Since the $3$ concurrent edges are along the coordinate axes,the origin is at $(0, 0, 0)$ and the opposite

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(B) Let the side length of the cube be $a$.Since the $3$ concurrent edges are along the coordinate axes,the origin is at $(0, 0, 0)$ and the opposite corner is at $(a, a, a)$.The direction ratios of the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are given by $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$.Substituting the coordinates,we get $(a - 0, a - 0, a - 0) = (a, a, a)$.Since direction ratios are proportional,we can divide by $a$ to get $(1, 1, 1)$.Thus,the direction ratios are $1, 1, 1$.

Class 12 Mathematics THREE DIMENSIONAL GEOMETRY System of co-ordinates, Direction cosines and direction ratios, Projection

Medium

The direction ratios of the diagonal of a cube that joins the origin to the opposite corner are (when the $3$ concurrent edges of the cube are the coordinate axes):

A
$2/\sqrt{3}, 2/\sqrt{3}, 2/\sqrt{3}$
B
$1, 1, 1$
C
$2, -2, 1$
D
$1, 2, 3$

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THREE DIMENSIONAL GEOMETRY

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System of co-ordinates, Direction cosines and direction ratios, Projection

If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then the value of ${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma + {\sin ^2}\delta$ is

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If $l, m, n$ are the direction cosines of a line which makes angles $\alpha, \beta$ and $\gamma$ with the coordinate axes $X, Y, Z$,respectively,then $l m+m n+n l$ takes the maximum value when

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If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of each of the $x$-axis and $y$-axis,then the angle that the line makes with the positive direction of the $z$-axis is

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If $\alpha, 2\alpha, 3\alpha$ are angles made by a ray with $OX, OY, OZ$ axes respectively,then all the possible values of $\alpha$ are:

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If $OP = 8$ and $\overrightarrow{OP}$ makes angles $45^\circ$ and $60^\circ$ with the $OX$-axis and $OY$-axis respectively,then $\overrightarrow{OP} = $

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The direction ratios of the diagonal of a cube that joins the origin to the opposite corner are (when the $3$ concurrent edges of the cube are the coordinate axes):

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If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then the value of ${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma + {\sin ^2}\delta$ is

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