Vector Algebra

Consider a$\Delta\,ABC$ where$A\left(1,3,2\right),B\left(-2,8,0\right)$ and$C\left(3,6,7\right)$. If the angle bisector of$\angle\,BAC$ meets the line $BC$at$D$, then the length of the projection of the vector$\vec{AD}$…

Let$x_{0}$ be the point of local maxima of…

Let $\vec{\mathrm{OA}}=\vec{a},\vec{\mathrm{OB}}=12\vec{a}+4\vec{b}$ and $\vec{\mathrm{OC}}=\vec{b}$, where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $\mathrm{OA}$ and $\mathrm{OC}$, then…

Let for a triangle $ABC$ $\vec{AB}=-2\hat{i}+\hat{j}+3\hat{k}$ $\vec{CB}=\alpha\,\hat{i}+\beta\,\hat{j}+\gamma\,\hat{k}$ $\vec{CA}=4\hat{i}+3\hat{j}+\delta\,\hat{k}$ If$\delta\,>0$and the area of the…

If the volume of parallelepiped formed by the vectors $\hat{i}+\lambda\,\hat{j}+\hat{k},\hat{j}+\lambda\,\hat{k}$ and $\lambda\,\hat{i}+\hat{k}$ is minimum, then $\lambda\,$ is equal to:

If the pointswith position vectors$\alpha\,\hat{i}+10\hat{j}+13\hat{k},6\hat{i}+11\hat{j}+11\hat{k},\frac{9}{2}\hat{i}+\beta\,\hat{j}-8\hat{k}$are collinear, then$\left(19\alpha\,-6\beta\,\right)^{2}$is equal to

Let $P Q R$ be a triangle such that $\overrightarrow{P Q}=-2 \hat{i}-\hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{PR}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}-4 \hat{\mathrm{k}}, a, b \in \mathbb{Z}$. Let S be the…

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda\gt0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the…

Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}.$If the vector $\vec{x}$ is perpendicular to $(3\hat{i}+2\hat{j}-\hat{k})$ and its…

Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b},\vec{a}\cdot\,\vec{c}=7$and $\vec{b}$ is perpendicular to $\vec{c},$where…

Let$\vec{a}=3\hat{i}+\hat{j}-2\hat{k},\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$and$\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$be three vectors. If a…

Let ABCDEF be a regular hexagon. If $\overrightarrow{\mathrm{AD}}=m \overrightarrow{\mathrm{BC}}$ and $\overrightarrow{\mathrm{CF}}=n \overrightarrow{\mathrm{AB}}$, then what is $m n$ equal to?