91Ó°ÊÓ

Prove that the perpendicular bisectors of the sides of a triangle are concurrent.

The line whose equation is \(\mathbf{r}=\lambda(2 \mathbf{i}-\mathbf{j}+\mathbf{k})\) has direction ratios (a) \(0,0,0\) (b) \(-2: 1:-1\) (c) \(2: 1: 1\) (d) \(\frac{1}{3}:-\frac{1}{6}: \frac{1}{6}\).

Prove that the diagonals of a rhombus intersect at right angles.

Does the line \(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\) lie in the plane \(A x+B y+C z+D=0 ?\) (a) \(A x_{1}+B y_{1}+C z_{1}+D=0\). (b) \(a A+b B+c C=0\). (c) The line and the plane are both 5 units from \(O\). (d) \(a: b: c=1: 2: 3\).

In a parallelogram \(\mathrm{ABCD} \mathrm{X}\) is the midpoint of \(\mathrm{AB}\) and the line DX cuts the diagonal \(\mathrm{AC}\) at \(\mathrm{P}\). Writing \(\overrightarrow{\mathrm{AB}}=\mathrm{a}, \overrightarrow{\mathrm{AD}}=\mathrm{b}, \quad \overrightarrow{\mathrm{AP}}=\lambda \overrightarrow{\mathrm{AC}} \quad\) and \(\overrightarrow{\mathrm{DP}}=\mu \overrightarrow{\mathrm{DX}}, \quad\) express \(\mathrm{AP}\) (a) in terms of \(\lambda, \mathbf{a}\) and \(\mathbf{b}\), (b) in terms of \(\mu, \mathbf{a}\) and \(\mathbf{b}\). Deduce that \(\mathrm{P}\) is a point of trisection of both \(\mathrm{AC}\) and \(\mathrm{DX}\).