Vinod Singh and Shweta Pawar Solutions for Chapter: Three Dimensional Geometry, Exercise 2: Critical Thinking

The angle between the two diagonals of a cube is

If D.R.'s of two lines are proportional to $2, 3, -6$ and $3, -4, 5$, then acute angle between these lines is

If the line joining the points $\left(-2,1,-8\right)$ and $\left(a,b,c\right)$ is parallel to the line whose direction ratios are $6, 2, 3$, then $a, b, c$ are

If the co-ordinates of the points $A, B, C, D$ be $\left(2, 3, -1\right), \left(3, 5, -3\right), \left(1, 2, 3\right)$ and $\left(3, 5, 7\right)$, respectively, then the projection of $AB$ on $CD$ is

The direction ratios of two lines are $a, b, c$ and $\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}$. The lines are

The direction cosines of two lines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$, then direction ratios of the line perpendicular to both lines are

The direction cosines of a line perpendicular to lines whose direction ratios are $2, 1, -3$ and $1, -2, 1$ are

The direction cosines of a line that are perpendicular to the lines with direction ratios $4, 1, 3$ and $2, -3, 1$ are