Three Dimensional Geometry - Algebra - Mathematics Previous Year Questions

A vector perpendicular to the plane containing the points \(A(1, - 1,2),B(2,0, - 1),C(0,2,1)\) is

The angle between the lines \(2x=3y=-z\) and \(6x=-y=-4z\) is

The equation of a plane passing through the line of intersection of the planes \(x+2y+3z=2,x-y+z=3\) and at a distance \(\frac{2}{\sqrt3}\) from the point \((3,1,-1)\) is

The point of intersection of the lines \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}\) and \({{x - 4} \over 5} = {{y - 1} \over 2} = z\) is

The line \({{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}\) is parallel to the plane

The point of intersection of the lines \({{x - 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}\) and \({{x -5} \over 2} = {{y - 2} \over 1} = z\) is

The line \(\frac{x-3}{4}=\frac{y-4}{5}=\frac{z-5}{6}\) is parallel to the plane

The angle between the lines \({{x + 4} \over 3} = {{y - 1} \over 5} = {{z + 3} \over 4}\) and \({{x + 1} \over 1} = {{y - 4} \over 1} = {{z - 5} \over 2}\) is

$$\mathrm{P} \text { is a point on the line segment joining the points }(3,2,-1) \text { and }(6,2,-2) \text {. If the } x \text { co ordinate of } \mathrm{P} \text { is } 5 \text {, then its } \mathrm{y} \text { coordinate is }$$

If the direction ratios of two lines are given by \(3 l m-4 l n+m n=0\) and \(l+2 m+3 n=0\), then the angle between the lines is

The distance of the point \((2,3,4)\) from the line \(1-x=\frac{y}{2}=\frac{1}{3}(1+z)\) is

The coordinates of the vertices of the triangle are \(A(-2,3,6), B(-4,4,9)\) and \(C(0,5,8)\). The direction cosines of the median \(\mathrm{BE}\) are

If the position vector of a point \(A\) is \(\vec{a}+2 \vec{b}\) and \(\vec{a}\) divides \(A B\) in the ratio \(2: 3\), then the position vector of \(B\) is

A line makes the same angle \(\theta\) with each of the \(x\) and \(z\)-axes. If the angle \(\beta\), which it makes with the \(y\)-axis is such that \(\sin ^2 \beta=3 \sin ^2 \theta\), then \(\cos ^2 \theta\) equals

The lines \(\vec{r}=(2 \hat{\jmath}-3 \hat{k})+\lambda(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})\) and \(\vec{r}=(2 \hat{\imath}+6 \hat{\jmath}+3 \hat{k})+\mu(2 \hat{\imath}+3 \hat{\jmath}+4 \hat{k})\) are

If the line \(\frac{1-x}{-3}=y=\frac{z+2}{2}\) is perpendicular to the line \(\frac{3 x-1}{2 b}=3-y=\frac{z-1}{a}\), then find the value of \(3 a+3 b\)

If the straight lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-t}\) and \(\frac{x-1}{t}=\frac{y-4}{2}=\frac{z-5}{1}\) are intersecting then \(t\) can have

The co-ordinate of the foot of the perpendicular from \(P(1,8,4)\) on the line joining \(R(0,-1,3)\) and \(Q(2,-3,-1)\) is

The measure of the angle between the lines \(x=k+1, \quad y=2 k-1, \quad z=2 k+3, \quad k \in R \quad\) and \(\quad \frac{x-1}{2}=\frac{y+1}{1}=\frac{z-1}{-2}\) is

\(P\) is a point on the line segment joining the points \((3,2,-1)\) and \((6,2,-2)\). If \(x\) coordinate of \(\mathrm{P}\) is 5, then its \(y\) co-ordinate is