3D Geometry

\({P_1}\) and \({P_2}\) are planes passing through origin. \({L_1}\) and \({L_2}\) are two line on \({P_1}\) and \({P_2}\) respectively such that their intersection is origin. Show that there exists points \(A, B, C,\) whose permutation $$A...

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 ...

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 \(ABC\) mee...

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 ............

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 \(R\) a...

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 + \alph...

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

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

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 points with position vectors \(60i+3j,\) \(40i-8j,\) \(ai-52j\) are collinear if

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 co...

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.