Three-Dimensional Geometry

The shortest distance between the lines $\dfrac{x+7}{-6}=\dfrac{y-6}7=z$ and $\dfrac{7-x}2=y-2=z-6$ is

If the system of linear equations.
$$8x+y+4z=-2$$
$$x+y+z=0$$
$$\lambda x-3y=\mu$$
has infinitely many solutions, then the distance of the point $\left(\lambda,\mu,-\dfrac12\right)$ from the plane $8x+y+4z+2=0$ is :

The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line $x+y-z=0=x-2y+3z-5$ is :

Let Q and R be two points on the line $\dfrac{x+1}2=\dfrac{y+2}3=\dfrac{z-1}2$ at a distance $\sqrt{26}$ from the point $P(4,2,7)$. Then the square of the area of the triangle PQR is ______.

A vector $\vec a$ is parallel to the line of intersection of the plane determined by the vectors $\hat i$, $\hat i+\hat j$ and the plane determined by the vectors $\hat i-\hat j$, $\hat i+\hat k$. The obtuse angle between $\vec a$ and the vector $\vec b=\hat i-2\hat j+2\hat k$ is

The largest value of $a$, for which the perpendicular distance of the plane containing the lines $\vec r=\left(\hat i+\hat j\right)+\lambda\left(\hat i+a\hat j-\hat k\right)$ and $\vec r=\left(\hat i+\hat j\right)+\mu\left(-\hat i+\hat j-a\hat k\right)$ from the point $(2,1,4)$ is $\sqrt3$, is ______.

The plane passing through the line $L:lx-y+3(1-l)z=1$, $x+2y-z=2$ and perpendicular to the plane $3x+2y+z=6$ is $3x-8y+7z=4$. If $\theta$ is the acute angle between the line L and the $y$-axis, then $415\cos^2\theta$ is equal to ______.

If the plane P passes through the intersection of two mutually perpendicular planes $2x+ky-5z=1$ and $3kx-ky+z=5$, $k<3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane P on the $y$-axis is

Let the line $\dfrac{x-3}7=\dfrac{y-2}{-1}=\dfrac{z-3}{-4}$ intersect the plane containing the lines $\dfrac{x-4}1=\dfrac{y+1}{-2}=\dfrac z1$ and $4ax-y+5z-7a=0=2x-5y-z-3$, $a\in\mathbf R$ at the point $P(\alpha,\beta,\gamma)$. Then the value of $\alpha+\beta+\gamma$ equals ______.

If the length of the perpendicular drawn from the point $P(a,4,2)$, $a>0$ on the line $\dfrac{x+1}2=\dfrac{y-3}3=\dfrac{z-1}{-1}$ is $2\sqrt6$ units and $Q(\alpha_1,\alpha_2,\alpha_3)$ is the image of the point P in this line, then $a+\displaystyle\sum_{i=1}^3\alpha_i$ is equal to :

If the line of intersection of the planes $ax+by=3$ and $ax+by+cz=0$, $a>0$ makes an angle 30° with the plane $y-z+2=0$, then the direction cosines of the line are :

The foot of the perpendicular from a point on the circle $x^2+y^2=1$, $z=0$ to the plane $2x+3y+z=6$ lies on which one of the following curves ?

Let $P(-2,-1,1)$ and $Q\left(\dfrac{56}{17},\dfrac{43}{17},\dfrac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2+\beta^2$ is equal to ______.

Let the lines $\dfrac{x-1}\lambda=\dfrac{y-2}1=\dfrac{z-3}2$ and $\dfrac{x+26}{-2}=\dfrac{y+18}3=\dfrac{z+28}\lambda$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?

A plane P is parallel to two lines whose direction ratios are −2, 1, −3 and −1, 2, −2 and it contains the point (2, 2, −2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts $\alpha$, $\beta$, $\gamma$. If V is the volume of the tetrahedron OABC, where O is the origin, and $p=\alpha+\beta+\gamma$, then the ordered pair (V, p) is equal to :

If the foot of the perpendicular from the point A(−1, 4, 3) on the plane P : $2x+my+nz=4$, is $\left(-2,\dfrac72,\dfrac32\right)$, then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, −1, −4, is equal to :

Let a line with direction ratios a, −4a, −7 be perpendicular to the lines with direction ratios 3, −1, 2b and b, a, −2. If the point of intersection of the line $\dfrac{x+1}{a^2+b^2}=\dfrac{y-2}{a^2-b^2}=\dfrac z1$ and the plane $x-y+z=0$ is (α, β, γ), then α + β + γ is equal to _______.

Let Q be the foot of perpendicular drawn from the point P (1, 2, 3) to the plane $x+2y+z=14$. If R is a point on the plane such that ∠PRQ = 60°, then the area of ΔPQR is equal to :

If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is :

Let $ax+by+cz+3=0$ be the plane which bisects the acute angle between the planes $2x-y-2z+3=0$ and $3x-2y+6z+8=0$. Then a + b + c is equal to