List of top Mathematics Questions on 3D Geometry asked in MET

The direction cosines of any normal to the xy-plane are

If \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are non-collinear vectors such that for some scalars \(x, y, z\), \(x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}\), then

A perpendicular is drawn from the point P(2, 4, -1) to the line \(\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}\). The equation of the perpendicular from P to the given line is

If \(\mathbf{a} = -\hat{\mathbf{i}} + 2\hat{\mathbf{j}} - \hat{\mathbf{k}}\), \(\mathbf{b} = \hat{\mathbf{i}} + \hat{\mathbf{j}} - 3\hat{\mathbf{k}}\) and \(\mathbf{c} = -4\hat{\mathbf{i}} - \hat{\mathbf{k}}\), then \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\) is

If P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5), then the projection of RS on PQ is

If a line makes \(\alpha, \beta, \gamma\) with the positive direction of x, y and z-axes respectively. Then, \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma\) is equal to

The projection of a line on a co-ordinate axes are 2, 3, 6. Then the length of the line is

If $a$, $b$, $c$ are the position vectors of $A$, $B$, $C$ respectively such that $3\mathbf{a} + 4\mathbf{b} - 7\mathbf{c} = \mathbf{0}$, then $C$ divides $AB$ in the ratio

The area of a parallelogram with diagonals $\mathbf{a} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k}$ and $\mathbf{b} = \mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$ is

If $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$, then $\mathbf{a}$ and $\mathbf{b}$ are

The work done by the force $4\hat{i} - 3\hat{j} + 2\hat{k}$ in moving a particle along a straight line from the point $(3, 2, -1)$ to $(2, -1, 4)$ is

If $\left(\dfrac{1}{2},\,\dfrac{1}{3},\,n\right)$ are the direction cosines of a line, then the value of $n$ is

If $\mathbf{a}= -\hat{i} + 2\hat{j} - \hat{k}$, $\mathbf{b} = \hat{i} + \hat{j} - 3\hat{k}$ and $\mathbf{c} = -4\hat{i} - \hat{k}$, then $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$ is

A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is

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

If $\vec{a}, \vec{b}, \vec{c}$ are any three mutually perpendicular vectors of equal magnitude a, then $|\vec{a} + \vec{b} + \vec{c}|$

If the position vectors of A, B, C are respectively $\hat{i} + \hat{j} - \hat{k}$, $2\hat{i} + 3\hat{j} + \hat{k}$ and $2\hat{i} - \hat{k}$, then the projection of $\overrightarrow{AB}$ on $\overrightarrow{BC}$ is equal to

$\vec{a} = \frac{1}{7}(2\hat{i} + 3\hat{j} + 6\hat{k})$, $\vec{b} = \frac{1}{7}(3\hat{i} - 2\hat{j} + \lambda \hat{k})$. If $\vec{a}$ and $\vec{b}$ are mutually perpendicular, then value of $\lambda$ is

If $\alpha, \beta, \gamma$ are the angles which a half ray makes with the positive direction of the axes, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

If the projection of $\overrightarrow{PQ}$ on OX, OY, OZ are respectively 12, 3 and 4, then the magnitude of $\overrightarrow{PQ}$ is