Vector Algebra

If $A B C D E F$ is a regular hexagon with $\mathbf{A B}=\mathbf{a}$ and $\mathbf{B C}=\mathbf{b}$, then $\mathbf{C E}$ equals to

Assertion $(A):$ If $\overset{\bar{}}{a},\overset{\bar{}}{b}$ are two non-collinear vectors then the vector component of $\overset{\bar{}}{b}$along the line perpendicular to $\overset{\bar{}}{a}$ is…

If the points $\mathbf{P}=\hat{i}+2 \hat{j}, \mathbf{Q}=4 \hat{i}+6 \hat{j}$, $\mathbf{R}=5 \hat{i}+7 \hat{j}, \mathbf{S}=a \hat{i}+b \hat{j}$ are the consecutive vertices of a parallelogram $P Q R S$, then

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three non-coplanar vectors, then match the items of List-I with those of List-II. The correct answer is A $\quad$ B $\quad$ B $\quad$ D

Let $\vec{a}, \vec{b}, \vec{c}$ be co-initial vectors and $\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}-\hat{\mathrm{k}}$. Let…

Let $\vec{a}=3 \hat{i}+4 \hat{j}-5 \hat{k}, \vec{b}=2 \hat{i}+\hat{j}-2 \hat{k}$. The projection of the sum of the vectors $\vec{a}, \vec{b}$ on the vector perpendicular to the plane of $\vec{a}, \vec{b}$ is

If $\overline{\mathrm{a}}$ is a unit vector, then $|\overline{\mathrm{a}} \times \overline{\mathrm{i}}|^2+|\overline{\mathrm{a}} \times \overline{\mathrm{j}}|^2+|\overline{\mathrm{a}} \times \overline{\mathrm{k}}|^2=$

If the position vectors of $A, B, C, D$ are $\bar{i}+2 \bar{j}+2 \bar{k}, 2 \bar{i}-\bar{j}, \bar{i}+\bar{j}+3 \bar{k}$ and $4 \overline{\mathrm{j}}+5 \overline{\mathrm{k}}$ respectively, then the quadrilateral ABCD is a

Let $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k}$ be two vectors, and $\overrightarrow{\mathrm{r}}$ be a vector along the vector $3 \overrightarrow{\mathrm{a}}-2…

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

Let $\overrightarrow{\mathrm{OA}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OB}}=-2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$ be the position vectors of two…

The number of vectors of unit length perpendicular to the two vectors $\mathbf{a}=(1,1,0)$ and $\mathbf{b}=(0,1,1)$ is