List of top Mathematics Questions on Three Dimensional Geometry asked in UP Board XII

Find the shortest distance between the lines \(\vec r=\hat i+\hat j-\lambda(2\hat i+\hat j+\hat k)\) and \(\vec r=2\hat i+\hat j+\hat k+\mu(3\hat i-5\hat j+2\hat k)\).

Find the angle between the lines \(\vec r = 3\hat i+2\hat j-4\hat k+\lambda(\hat i+2\hat j+2\hat k)\) and \(\vec r = 5\hat i-2\hat j+\mu(3\hat i+2\hat j+6\hat k)\).

Find the direction cosines of the \(x\), \(y\) and \(z\) axes.

Find the Cartesian equation of the line which passes through the point \((-1,4,-5)\) and is parallel to \(\dfrac{x+3}{3}=\dfrac{y-4}{5}=\dfrac{z-8}{6}\).

Find the shortest distance between the lines \(\vec r=(\hat i+2\hat j+\hat k)+\lambda(\hat i-\hat j+\hat k)\) and \(\vec r=(2\hat i-\hat j-\hat k)+\mu(2\hat i+\hat j+2\hat k)\).

Find the vector and Cartesian equations of a line which passes through the point \((1,2,3)\) and is parallel to the vector \(2\hat i+3\hat j+2\hat k\).

Direction ratios of a line are 2, \(-1\) and \(-2\). Find the direction cosines of the line.

Direction cosines of the \(x\)-axis are:

Find the shortest distance between the lines \(\vec r=\hat i+2\hat j-4\hat k+\lambda(2\hat i+3\hat j+6\hat k)\) and \(\vec r=3\hat i+3\hat j-5\hat k+\mu(2\hat i+3\hat j+6\hat k)\).

Show that the line passing through the points \((1,-1,2)\) and \((3,4,-2)\) is perpendicular to the line passing through the points \((0,3,2)\) and \((3,5,6)\).

Show that the points \(A(2,3,-4)\), \(B(1,-2,3)\) and \(C(3,8,-11)\) are collinear.

Find the minimum distance between the lines \(\vec r=\hat i+\hat j+\lambda(2\hat i-\hat j+\hat k)\) and \(\vec r=2\hat i+\hat j-\hat k+\mu(3\hat i-5\hat j+2\hat k)\).

If the direction ratios of a line are \(3,-2,-6\), then find its direction cosines.

If the lines \(\dfrac{x-1}{-3}=\dfrac{y-2}{2k}=\dfrac{z-3}{2}\) and \(\dfrac{x-1}{3k}=\dfrac{y-1}{1}=\dfrac{z-6}{-5}\) are mutually perpendicular, find the value of \(k\).

Find the Cartesian equation of the line parallel to the vector \(3\hat i+2\hat j-8\hat k\) and passing through the point \((5,2,-4)\).

If a line makes \(90^\circ\), \(60^\circ\) and \(30^\circ\) with the \(x\), \(y\) and \(z\)-axes respectively in the positive direction, then find its direction cosines.