Vinod Singh and Shweta Pawar Solutions for Chapter: Vectors, Exercise 3: Competitive Thinking

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-collinear vectors and $x\overrightarrow{a}+y\overrightarrow{b}=0,$ then

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

$\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-collinear vectors, then $x\overrightarrow{a}+y\overrightarrow{b}$ (where $x$ and $y$ are scalars) represents a vector which is

If $\overrightarrow{a}=\hat{i}+\hat{j}-2\hat{k}, \overrightarrow{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{k}$ and $\overrightarrow{c}=m\overrightarrow{a}+n\overrightarrow{b}$ then $m+n=$

The position vector of the point which divides the join of the points $2\overrightarrow{a}-3\overrightarrow{b}$ and $3\overrightarrow{a}-2\overrightarrow{b}$internally in the ratio $2:3$ is

Let $A\left(1,-1,2\right)$ and $B\left(2,3,-1\right)$ be two points. If a point $P$ divides $AB$ internally in the ratio $2:3,$ then the position vector of $P$ is

If the orthocentre and centroid of a triangle are $\left(-3,5,2\right)$ and $\left(3,3,4\right)$, respectively, then its circumcentre is

If the four points with position vectors $-2\hat{i}+\hat{j}+k, \hat{i}+\hat{j}+\hat{k}, \hat{j}-\hat{k}$ and $\lambda \hat{j}+\hat{k}$are coplanar, then  $\lambda =$