3D Geometry

Find the equation of plane passing through $(1, 1, 1)$ & parallel to the lines ${L_1},{L_2}$ having direction ratios $(1,0,-1),(1,-1,0).$ Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.

The value of $k$ such that ${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$ lies in the plane $2x -4y +z = 7,$ is

(i) Find the equation of the plane passing through the points $(2, 1, 0), (5, 0, 1)$ and $(4, 1, 1).$
(ii) If $P$ is the point $(2, 1, 6)$ then find the point $Q$ such that $PQ$ is perpendicular to the plane in (i) and the mid point of $PQ$ lies on it.

The position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ are $\widehat i + \widehat j + \widehat k,\,\widehat i$ and $3\widehat i\,,$ respectively. The altitude from vertex $D$ to the opposite face $ABC$ meets the median line through $A$ of the triangle $ABC$ at a point $E.$ If the length of the side $AD$ is $4$ and the volume of the tetrahedron is ${{2\sqrt 2 } \over 3},$ find the position vector of the point $E$ for all its possible positions.

Let $\overrightarrow p $ and $\overrightarrow q $ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $\left| {\overrightarrow p } \right| = p,\left| {\overrightarrow q } \right| = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2:3$ respectively. If $OR$ and $OS$ are perpendicular then

Let $\alpha ,\beta ,\gamma $ be distinct real numbers. The points with position
vectors $\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alpha \widehat j + \beta \widehat k$

A unit vector perpendicular to the plane determined by the points $P\left( {1, - 1,2} \right)Q\left( {2,0, - 1} \right)$ and $R\left( {0,2,1} \right)$ is ............

The points with position vectors $60i+3j,$ $40i-8j,$ $ai-52j$ are collinear if

The volume of the parallelopiped whose sides are given by
$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC} = 3i - k,$ is

A vector $\overrightarrow A $ has components ${A_1},{A_2},{A_3}$ in a right -handed rectangular Cartesian coordinate system $oxyz.$ The coordinate system is rotated about the $x$-axis through an angle ${\pi \over 2}.$ Find the components of $A$ in the new coordinate system in terms of ${A_1},{A_2},{A_3}.$

The unit vector perpendicular to the plane determined by $P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)$ and $R\left( {0,2,1} \right)$ is ...........

The area of the triangle whose vertices are $A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)$ is ..........

From a point $O$ inside a triangle $ABC,$ perpendiculars $OD$, $OE, OF$ are drawn to the sides $BC, CA, AB$ respectively. Prove that the perpendiculars from $A, B, C$ to the sides $EF, FD, DE$ are concurrent.