Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

The direction cosines of the $x-$axis are _____.

Find the position vector of the point which divides the line segment joining the points with position vectors $2\hat{i}-4\hat{j}+3\hat{k}, -4\hat{i}+5\hat{j}-6\hat{k}$ in the ratio $2:1$. 

Find the position vector of a point $R$ which divides the line segment joining $P\left(3\overrightarrow{a}-2\overrightarrow{b}\right)$ and $Q\left(\overrightarrow{a}+\overrightarrow{b}\right)$ in the ratio $2:1$ externally.

If a vector makes angles $\alpha , \beta , \gamma$ with the co-ordinate axes, then show that, ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$.

The projections of a vector $\overrightarrow{a}$ on the axes are $3, 4$ and $12$; find the length of $\overrightarrow{a}$ and the direction cosines of $\overrightarrow{a}$.

If the position vectors of two points $A$ and $B$ are $3\hat{i}+5\hat{j}+\hat{k}$ and $5\hat{i}+11\hat{j}+4\hat{k}$, then find the projections of $\overrightarrow{AB}$ on the co-ordinate axes and the direction cosines of $\overrightarrow{AB}$.

If the diagonals of a quadrilateral bisect each other, then prove by vector method that the quadrilateral is a parallelogram.

$AD$ is a median of the triangle $ABC. E$ is the midpoint of the line segment $AD. BE$ is joined and produced to meet $AC$ at $F$. Prove by vector method that $AF=\frac{1}{3}AC$ .