G Tewani Solutions for Chapter: Introduction to Vectors, Exercise 1: DPP

The orthocentre of an equilateral triangle $ABC$ is the origin $O$. If $\overrightarrow{OA}=\overrightarrow{a}$, $\overrightarrow{OB}=\overrightarrow{b}$ and $\overrightarrow{OC}=\overrightarrow{c}$, then $\overrightarrow{AB}+2\overrightarrow{BC}+3\overrightarrow{CA}$ is equal to

If the position vectors of $P$ and $Q$ are $\widehat{i}+2\widehat{j}-7\widehat{k}$ and $5\widehat{i}-3\widehat{j}+4\widehat{k}$, respectively, then the cosine of the angle between $\overrightarrow{PQ}$ and $z$-axis is

The non-zero vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ are related by $\overrightarrow{a}=8\overrightarrow{b}$ and $\overrightarrow{c}=-7\overrightarrow{b}$, the angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is

The unit vector bisecting $\overrightarrow{OY}$ (representing $Y$-axis) and $\overrightarrow{OZ}$ (representing $Z$- axis) is

A unit tangent vector at $t=2$ on the curve $x=t^2+2$, $y=4t-5$ and $z=2t^2-6t$ is

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the position vectors of $A$ and $B$, respectively, then the position vector of a point $C$ is $\overrightarrow{AB}$, produced such that $\overrightarrow{AC}=2015\overrightarrow{AB}$.

Let $\overrightarrow{a}=\left(1, 1, -1\right)$, $\overrightarrow{b}=\left(5, -3, -3\right)$ and $\overrightarrow{c}=\left(3, -1, 2\right)$. If $\overrightarrow{r}$ is collinear with $\overrightarrow{c}$ and has length of $\frac{\left|\overrightarrow{a}+\overrightarrow{b}\right|}{2}$, then $\overrightarrow{r}$equals

A line passes through the points whose position vectors are $\widehat{i}+\widehat{j}-2\widehat{k}$ and $\widehat{i}-3\widehat{j}+\widehat{k}$. The position vector of a point on the line at unit distance from the first point can be