R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 6: EXERCISE

The position vector of a point $R$ which divides the line segment joining points: $P\left(\hat{i}+2\hat{j}+\hat{k}\right)$ and $Q\left(-\hat{i}+\hat{j}+\hat{k}\right)$ internally is $k(-\hat{i}+4\hat{j}+3\hat{k})$, then the value of $k$ is

The position vector of a point $R$ which divides the line segment joining points(externally): $P\left(\hat{i}+2\hat{j}+\hat{k}\right)$ and $Q\left(-\hat{i}+\hat{j}+\hat{k}\right)$ is $p\hat{i}+\hat{k}$, then the value of $p$ is

The position vector of the mid-point of the vector joining the points $P\left(2\hat{i}-3\hat{j}+4\hat{k}\right)$ and $Q\left(4\hat{i}+\hat{j}-2\hat{k}\right)$ is $p\hat{i}-\hat{j}+\hat{k}$, then the value of $p$ is

The unit vector in the direction of vector $\overrightarrow{PQ},$ where $P$ and $Q$ are the points $(1, 2, 3)$and $(4, 5, 6)$ is $\frac{1}{\sqrt{k}}\left(\hat{i}+\hat{j}+\hat{k}\right)$, then the value of $k$ is

The position vector of the mid-point of the vector joining the points $P(2, 3, 4)$ and $Q(4, 1, -2)$ is $p\hat{i}+2\hat{j}+\hat{k}$, then the value of $k$ is

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=4\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{c}=\hat{i}-2\hat{j}+\hat{k},$ and a vector of magnitude $6$ units which is parallel to the vector $2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}$ is $2\hat{i}-p\hat{j}+4\hat{k}$, then the value of $p$ is

Find a vector of magnitude of $5$ units parallel to the resultant of the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}+\hat{k}.$

The two vectors $\hat{j}+\hat{k}$ and $3\hat{i}-\hat{j}+4\hat{k}$ represents the sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ respectively of a triangle $ABC$. Find the length of the median through $A.$