List of top Mathematics Questions on introduction to three dimensional geometry

The shortest distance from the point $(-10, 10, -10)$ to the $z$-axis, is
Let \(O\) be the origin and \(\vec r\) be the position vector of a point \(P\). If \(\overline{OP}\) makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\) with \(\overline{i}\) and \(\overline{j}\) respectively, then a vector along \(\overline{OP}\) with magnitude 2 is
The ratio in which the $xy$-plane divides the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is:
Find the distance of the point \( (1,-2,3) \) from the \(yz\)-plane.
What is the distance of the point \((1,2,3)\) from the \(yz\)-plane?
If the sum of the squares of the distances of a point \(P(x, y, z)\) from the three co-ordinate axes is \(324\), then the distance of point \(P\) from the origin is ....
The projections of a line segment on the coordinate axes are \(5,6,8\). Then the length of the line segment is
Let \( \alpha, \beta \) and \( \gamma \) be the angles made by a straight line with the x-axis, y-axis and z-axis respectively. If \( \cos\alpha + \cos\beta + \cos\gamma = \frac{5}{3} \), then the value of \( \cos\alpha \cos\beta + \cos\beta \cos\gamma + \cos\gamma \cos\alpha \) is equal to
If two vertices of a triangle are $A(3,1,4)$ and $B(-4,5,-3)$ and the centroid of the triangle is $G(-1,2,1),$ then the third vertex C of the triangle is
Let a circle of radius 4 be concentric to the ellipse \( 15x^2 + 19y^2 = 285 \). Then the common tangents are inclined to the minor axis of the ellipse at the angle:
If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is
If the coordinates of the vertices of a triangle \(ABC\) are \[ A(7,6,4),\quad B(5,4,6),\quad C(3,2,0) \] and the bisector of \(\angle BAC\) meets the side \(BC\) at \(D\), then the coordinates of \(D\) are:
If the direction cosines of a line satisfy the relations \[ l-m+n=0 \] and \[ lm+mn-4nl=0, \] then the direction cosines of the line are:
Let the abscissae of the two points \(P\) and \(Q\) be the roots of \(2x^2 – rx + p = 0\) and the ordinates of \(P\) and \(Q\) be the roots of \(x^2 – sx – q = 0\). If the equation of the circle described on \(PQ\) as diameter is \(2(x^2 + y^2) – 11x – 14y – 22 = 0\), then \(2r + s – 2q + p\) is equal to _________.
If the points $P(4,5,x)$, $Q(3,y,4)$ and $R(5,8,0)$ are collinear, then the value of $x+y$ is
If $G(4,3,3)$ is the centroid of the triangle $ABC$ whose vertices are $A(a,3,1)$, $B(4,5,b)$ and $C(6,c,5)$, then the values of $a$, $b$, $c$ are
The graph of equation \(y^2 + z^2 = 0\) in three dimensional space is
If the equation of the sphere through the circle $x^2 + y^2 + z^2 = 5$, $2x + 3y + 4z = 5$ and through the origin is $x^2 + y^2 + z^2 - 2x - 3y - 4z + C = 0$, then the value of $C$ is
A line makes the same angle θ with each of the X and Z-axes. If the angle β, which it makes with Y-axis, is such that sin²β=3sin²θ, then cos²θ equals
The ratio in which the line joining the points \( (2,1,5) \) and \( (3,4,3) \) is divided by the plane \( x + y - z = \dfrac{1}{2} \) is:
If the line through the points A(k,1,-1) and B(2k,0,2) is perpendicular to the line through the points B and C(2+2k,k,1), then the value of k is
A line makes the same angle \(\theta\) with each of the X and Z-axes. If the angle \(\beta\) which it makes with Y-axis is such that \(\sin^2\beta=3\sin^2\theta\), then \(\cos^2\theta\) equals
If $\triangle ABC$ is right angled at A, where $A\equiv(4,2,x)$, $B\equiv(3,1,8)$ and $C\equiv(2,-1,2)$, then the value of $x$ is
The image of the line $\frac{x-2}{3}=\frac{y+1}{4} = \frac{z-2}{12}$ in the plane $2x - y + z + 3 = 0$ is the line
The radius of the sphere $x^2 + y^2 + z^2 = 12x + 4y + 3z$ is