Vinod Singh and Shweta Pawar Solutions for Chapter: Vectors, Exercise 4: Evaluation Test

If the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-coplanar and $l, m, n$ are distinct scalars such that $\left[l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c} l\overrightarrow{b}+m\overrightarrow{c}+n\overrightarrow{a} l\overrightarrow{c}+m\overrightarrow{a}+n\overrightarrow{b}\right]=0,$ then

$P$ is any point on the circumference of the circumcircle of $∆ABC. H$ is the orthocentre, $M$ is the midpoint of $PH$ and $D$ is the midpoint of $BC$, then

The three vectors $10\hat{i}+13\hat{j}+16\hat{k}, 30\hat{i}+33\hat{j}+36\hat{k}$ and $47\hat{i}+50\hat{j}+53\hat{k}$ are

If the volume of parallelepiped whose concurrent edges are $3\hat{i}-\hat{j}+4\hat{k}, 2\hat{i}+\lambda \hat{j}-\hat{k}$ and $-5\hat{i}+2\hat{j}+\lambda \hat{k}$ is $110 \mathrm{cu}.\mathrm{units}$, then the value of $\lambda$ is

If the vectors $5\hat{i}-x\hat{j}+3\hat{k}$ and $-3i+2\hat{j}-y\hat{k}$ are parallel, the values of $x$ and $y$, respectively, are

If the position vector of $P$ is $3\overrightarrow{p}+\overrightarrow{q}$ and $\overrightarrow{p}$ divides $PQ$ internally in the ratio $3:4$, the position vector of $Q$ is 

In $∆\mathrm{ABC}$ the mid points of the sides $\mathrm{AB}$, $\mathrm{BC}$ and $\mathrm{CA}$ are, respectively, $\left(\mathrm{l},0,0\right), \left(0,\mathrm{m},0\right), \left(0,0,\mathrm{n}\right)$. Then $\frac{{\mathrm{AB}}^2+{\mathrm{BC}}^2+{\mathrm{CA}}^2}{{\mathrm{l}}^2+{\mathrm{m}}^2+{\mathrm{n}}^2}=$

The co-ordinates of the foot of perpendicular drawn from the point $A\left(1,0,3\right)$ to the line joining $B\left(4,7,1\right)$ and $C\left(3,5,3\right)$ is