Let the position vectors of two points and be and , respectively. Let and be two points such that the direction ratios of lines and are and , respectively. Let lines and intersect at . If the vector is perpendicular to both and and the length of vector is units, then the modulus of a position vector of is :
The angle between the straight lines, whose direction cosines are given by the equations and , is:
For real numbers and if the point of intersection of the straight lines and lies on the plane then is equal to :
Let be the angle between the lines whose direction cosines satisfy the equations and . Then the value of is :
If the length of the perpendicular drawn from the point on the line is units and is the image of the point in this line, then is equal to
Let and be the vertices of the rhombus . If the direction ratios of the diagonal are , where both and $\beta$ are integers of minimum absolute values, then is equal to
The line passes through the point and is perpendicular to the plane . Then the shortest distance between the line and the line is:
Let be the foot of perpendicular from the point on the line passing through the points and . Then the distance of from the plane is
The shortest distance between the lines and is
Let a line pass through the origin and be perpendicular to the lines and
. If is the point of intersection of and , and , is the foot of perpendicular from on , then is equal to ________.
If the line $\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$, then $4 \lambda+9 \mu$ is equal to :
The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is
Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1,7,6)$ on the line passing through the points $(2,-5,11)$ and $(-6,7,-5)$. Then the length of the line segment $P Q$ is equal to ________
Let $P(x, y, z)$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P=\gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is
Let the line $\mathrm{L}$ intersect the lines $x-2=-y=z-1,2(x+1)=2(y-1)=z+1$ and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following points lies on L?
If the mirror image of the point in the line is then is:
Let , and be three lines such that is perpendicular to and is perpendicular to both and . Then the point which lies on is
Let the line of the shortest distance between the lines and intersect and at and respectively. If is the midpoint of the line segment , then is equal to___________
Let be the points on the line which are at a distance of units from the point . If the centroid of the triangle is then is:
Let and be the feet of perpendiculars from the point on the lines and respectively. If is a right angle, then is equal to ________
Consider the line $L$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from the line $\mathrm{L}$ along the line $\frac{3 x-11}{2}=\frac{3 y-11}{1}=\frac{3 z-19}{2}$ is equal to
The square of the distance of the image of the point $(6,1,5)$ in the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$, from the origin is _________
If is the shortest distance between the lines and is the shortest distance between the lines , then the value of is :
The lines and intersect at the point . If the distance of from the line is , then is equal to _____.
If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k}$, and $\int_0^{\mathrm{k}}\left[x^2\right] \mathrm{d} x=\alpha-\sqrt{\alpha}$, where $[x]$ denotes the greatest integer function, then $6 \alpha^3$ is equal to________