Three Dimensional Geometry

The Cartesian equation of a line is \(3 x+1=6 y-2=1-z\), then its vector equation is

The area of triangle with vertices \((1,2,0),(1,0, a)\) and \((0,3,1)\) is \(\sqrt{6}\) sq. units, then the values of '\(a\)' are

If \(\mathrm{G}(4,3,3)\) is the centroid of the triangle \(\mathrm{ABC}\) whose vertices are \(\mathrm{A}(\mathrm{a}, 3,1), \mathrm{B}(4,5, \mathrm{~b})\) and \(C(6, c, 5)\), then the values of \(a, b, c\) are

A line drawn from a point \(A(-2,-2,3)\) and parallel to the line \(\frac{x}{-2}=\frac{y}{2}=\frac{z}{-1}\) meets the \(\mathrm{YOZ}\) plane in point \(\mathrm{P}\), then the co-ordinates of the point \(\mathrm{P}\) are

The line \(\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}\) lies in the plane \(x+3 y-\alpha z+\beta=0\), then value of \(\alpha \beta\) is

The d.r.s. of the normal to the plane passing through the origin and the line of intersection of the planes \(x+2 y+3 z=4\) and \(4 x+3 y+2 z=1\) are

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

The equation of the plane passing through \((-2,2,2)\) and \((2,-2,-2)\) and perpendicular to the plane \(9 x-13 y-3 z=0\) is

The Cartesian equation of the plane passing through the point A(7, 8, 6) and parallel to the XY plane is

If the lines $\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles, then $\lambda=$

The area of the parallelogram with vertices A(1, 2, 3), B(1, 3, a), C(3, 8, 6) and D(3, 7, 3) is \(\sqrt{265}\) sq. units, then a =

The equation of a line passing through \((3,-1,2)\) and perpendicular to the lines \(\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-2 \hat{j}+\hat{k})\) and \(\bar{r}=(2 \hat{i}+\hat{j}-3 \hat{k})+\mu(\hat{i}-2 \hat{j}+2 \hat{k})\) is

The equation of the plane containing the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\) and the point \((0,7,-7)\) is

The co-ordinates of the point \(\mathrm{P} \equiv(1,2,3)\) and \(\mathrm{O} \equiv(0,0,0)\), then the direction cosines of \(\overline{\mathrm{OP}}\) are

Equation of the plane passing through the point (2, 0, 5) and parallel to the vectors \(\widehat i - \widehat j + \widehat k\) and \(3\widehat i + 2\widehat j - \widehat k\) is

The direction cosines \(\ell, \mathrm{m}, \mathrm{n}\) of the line \(\frac{\mathrm{x}+2}{2}=\frac{2 \mathrm{y}-5}{3} ; \mathrm{z}=-1\) are

If the line \(\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6}\) lies in the plane \(3 x-14 y+6 z+49=0\), then the value of \(m\) is

The angle between a line with direction ratios 2, 2, 1 and a line joining (3, 1, 4) and (7, 2, 12) is

The parametric equations of a line passing through the points \(\mathrm{A}(3,4,-7)\) and \(\mathrm{B}(1,-1,6)\) are

If the lines \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}\) intersect, then the values of \(k\) is

The Cartesian equation of the plane passing through the point \((0,7,-7)\) and containing the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\) is

The vector equation of the line whose Cartesian equations are y = 2 and 4x \(-\) 3z + 5 = 0 is

The equation of the plane that contains the line of intersection of the planes. \(x+2 y+3 z-4=0\) and \(2 x+y-z+5=0\) and is perpendicular to the plane \(5 x+3 y-6 z+8=0\) is

The Cartesian equation of the plane \(\overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})\) is

The Cartesian equation of the line passing through the points A(2, 2, 1) and B(1, 3, 0) is

The equation of the line passing through $(1,2,3)$ and perpendicular to the lines $x-1=\frac{y+2}{2}=\frac{z+4}{4}$ and $\frac{x-1}{2}=\frac{y-2}{2}=z+3$ is

The equation of a plane containing the point $(1,-1,2)$ and perpendicular to the planes $2 x+3 y-2 z=5$ and $x+2 y-3 z=8$ is

The equations of planes parallel to the plane $x+2 y+2 z+8=0$, which are at a distance of 2 units from the point $(1,1,2)$ are

A line makes angles $\alpha, \beta, \gamma$ with the co-ordinate axes and $\alpha+\beta=90^{\circ}$, then $\gamma=$

The point $P$ lies on the line $A, B$ where $A=(2,4,5)$ and $B \equiv(1,2,3)$. If $z$ co-ordinate of point $P$ is 3 , the its $y$ co-ordinate is

If the lines given by \(\frac{x-1}{2 \lambda}=\frac{y-1}{-5}=\frac{z-1}{2}\) and \(\frac{x+2}{\lambda}=\frac{y+3}{\lambda}=\frac{z+5}{1}\) are parallel, then the value of \(\lambda\) is

The direction cosines of a line which is perpendicular to lines whose direction ratios are \(3,-2,4\) and \(1,3,-2\) are

The angle between the line \(r =(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}})\) and the plane \(\mathbf{r} \cdot(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})=8\) is

If the foot of perpendicular drawn from the origin to the plane is \((3,2,1)\), then the equation of plane is

The direction co-sines of the line which bisects the angle between positive direction of \(Y\) and \(Z\) axes are

If the plane \(2 x+3 y+5 z=1\) intersects the co-ordinate axes at the points \(A, B, C\), then the centroid of \(\triangle A B C\) is

If the line \(r=(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(r \cdot(3 \hat{i}-2 \hat{\mathbf{j}}+m \hat{\mathbf{k}})=10\), then the va...

The cosine of the angle included between the lines \(\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\) and $$\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}+3...

The angle between the lines \(\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}\) and \(\frac{x-2}{2}=\frac{y+1}{2}=\frac{z-4}{1}\) is

The points \(A(-a,-b), B(0,0), C(a, b)\) and \(D\left(a^2, a b\right)\) are

Which of the following can not be the direction cosines of a line?

If line $\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}$ and $\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}$ are perpendicular to each other then $\lambda=$ ............

The direction ratios of the normal to the plane passing through origin and the line of intersection of the planes $x+2 y+3 z=4$ and $4 x+3 y+2 z=1$ are $\ldots \ldots$

The vector equation of the plane $\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})$ in scalar product form is $\mathbf{r} \cdot(3 \hat{\mathbf{i}}+2 \hat{...

If the line passes through the points $P(6,-1,2), Q(8,-7,2 \lambda)$ and $R(5,2,4)$ then value of $\lambda$ is ...........

The angle between lines $\frac{x-2}{2}=\frac{y-3}{-2}=\frac{z-5}{1}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-5}{2}$ is ............

The co-ordinates of the foot of perpendicular drawn from origin to the plane $2 x-y+5 z-3=0$ are $\ldots \ldots$

If $G(3,-5, r)$ is centroid of triangle $A B C$ where $A(7,-8,1), B(p, q, 5)$ and $C(q+1,5 p, 0)$ are vertices of a triangle then values of $p, q, r$ are respectively ......

The equation of the plane passing through the point $(-1,2,1)$ and perpendicular to the line joining the points $(-3,1,2)$ and $(2,3,4)$ is

If $P(6,10,10), Q(1,0,-5), R(6,-10, \lambda)$ are vertices of a triangle right angled at $Q$, then value of $\lambda$ is ............