Area of Parallelogram and Triangle

A vector of magnitude $9$ and perpendicular to both the vectors $4\hat{\mathrm{i}}-\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ & $-2\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ is:

If $2\overrightarrow{a}+5\overrightarrow{b}-9\overrightarrow{c}=\overrightarrow{0}$, then $\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \left[\left(2\overrightarrow{b}\times 3\overrightarrow{c}\right)\times \left(3\overrightarrow{c}\times 6\overrightarrow{a}\right)\right]$ is equal to

The area of the parallelogram whose adjacent sides are determined by the vectors $\overrightarrow{a}=\hat{ı}-\hat{ȷ}+3\hat{k}, \overrightarrow{b}=2\hat{ı}-7\hat{ȷ}+\hat{k}$ is $k\sqrt{2} \mathrm{sq}.\mathrm{units}$, then find the value of $k$.

If two sides of a triangle are represented by vectors $\hat{i}+2\hat{j}+2\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$, then the area of triangle is

If the two sides of the triangle are given by $\hat{i}+2\hat{j}+2\hat{k}$ and $3\hat{i}-2\hat{j}+1\hat{k}$ then find the area of the triangle.

Find the value of $k$. If the area of the triangle whose two adjacent sides are determined by the vectors $\overrightarrow{a}=-2\hat{i}-5\hat{k}\text{ and }\overrightarrow{b}=\hat{i}-2\hat{j}-\hat{k}$ is $\frac{\sqrt{k}}{2}$ sq, units

Find the value of $m$, if area of the parallelogram whose adjacent sides are represented by the vectors: $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k}\text{ and }\overrightarrow{b}=\hat{i}-\hat{j}$ is equal to $3\sqrt{m}$ square units.

Find the value of $k$, if the area of the parallelogram whose adjacent sides are represented by the vectors: $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}\text{ and }\overrightarrow{b}=-3\hat{i}-2\hat{j}+\hat{k}$ is $6\sqrt{k}$sq. units.

If $\overrightarrow{a}, \overrightarrow{b},\overrightarrow{ c}$ are the vertices of a triangle and area of triangle is given as $k[\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}]$, then the value of $k$ is

Consider a triangle $ABC$, with an internal point ' $O$ ' satisfying the relation $3\overrightarrow{OA}+2\overrightarrow{OB}+\overrightarrow{OC}=\overrightarrow{0}$ and  $\frac{r_1}{r_2}=\frac{Area\left(\Delta ABC\right)}{Area\left(\Delta AOC\right)},$ then $\left[\frac{4r_1+r_2}{r_2}\right]$ is equal to ($r_1$ and $r_2$ are co-prime numbers)

Let $\overrightarrow{p}$ and $\overrightarrow{q}$ be two non-zero, non-collinear vectors $\left(\right|\overrightarrow{p}|=1)$ such that vectors $3(\overrightarrow{p}\times \overrightarrow{q})$ and $2(\overrightarrow{\mathrm{q}}-(\overrightarrow{\mathrm{p}}·\overrightarrow{\mathrm{q}}\left)\overrightarrow{\mathrm{p}}\right)$ represents two sides of a triangle. If the area of the triangle is $\frac{3}{4}\left(|\overrightarrow{\mathrm{q}}{|}^4+4\right)$. Then $7|\overrightarrow{\mathrm{q}}{|}^2$ is equal to

The plane containing the two straight lines $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ and $\overrightarrow{r}=\overrightarrow{b}+\mu \overrightarrow{a}$ are.

Let the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be the position vectors of the vertices $P, Q, R$, respectively, of a triangle. Which of the following represents the area of the triangle?

The area of the triangle having the points $A(1,1,1), B(1,2,3)$ and $C(2,3,1)$ as its vertices is 

Let $O$ be the interior point of $△ABC$ such that $2\overrightarrow{OA}+3\overrightarrow{OB}+6\overrightarrow{OC}=0$ where $O$ is origin. If $\frac{\text{Area}\text{of}\left(\text{ΔABC}\right)}{\text{Area}\text{of}\left(\text{ΔAOB}\right)}=\frac{m}{n}$ , where $m$ and $n$ are relatively prime, then $\left(m-n\right)$ is equal to _______

Let $\overrightarrow{OA}$= $\overrightarrow{a}$, $\overrightarrow{OB}$= 2 $\overrightarrow{a}$+ 10 $\overrightarrow{b}$, $\overrightarrow{OC}$= $\overrightarrow{b}$where O, A, C are non collinear points. Let $l$ denote the area of the quadrilateral OABC. Let m denotes the area of parallelogram with $\overrightarrow{OA}$and $\overrightarrow{OC}$ as adjacent sides. If $l$ = 2λm then find the value of λ.

A triangle has vertices $\left(1, 1, 1\right), \left(2, 2, 2\right), \left(1, 1, y\right)$ and its area equals to $\mathrm{cosec}\left(\frac{\mathrm{\pi }}{4}\right)\text{ sq. units}$. The sum of the possible values of $\text{'}y\text{'}$ is

If $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=0$ then