Vector Algebra

The area of the rectangle having vertices $\mathrm{P}, \quad \mathrm{Q}, \quad \mathrm{R}, \quad \mathrm{S}$ with position vectors $-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \ha...

Let $\bar{a}=2 \hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-\hat{k}$ and vector $\bar{c}$ be coplanar. If $\bar{c}$ is perpendicular to $\bar{a}$, then $\bar{c}$ is

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=1$. If...

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0},|\overline{\mathrm{a}}|=3,|\overline{\mathrm{~b}}|=4,|\overline{\mathrm{c}}|=5$, then $\overline{\mat...

If $\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\bar{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$, then the value of $(2 \overline{\mathrm{a}}-\overline{\mathrm{b}}) \cdot((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \tim...

If $\bar{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \bar{b}=\hat{i}-2 \hat{j}-2 \hat{k}, \bar{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$ and if $\overline{\mathrm{d}}$ is vector perpendicular to both $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}, \overli...

Let $\bar{a}=\hat{i}+\hat{j}, \bar{b}=2 \hat{i}-\hat{k}, \bar{c}=3 \hat{i}-\hat{j}+\hat{k}$, then vector $\overline{\mathrm{p}}$ satisfying $\overline{\mathrm{p}} \cdot \overline{\mathrm{a}}=0$ and $\overline{\mathrm{p}} \times \overline{\m...

If the vectors $m \hat{i}+m \hat{j}+n \hat{k}, \hat{i}+\hat{k}, n \hat{i}+n \hat{j}+p \hat{k}$ lie in a plane then…

$\bar{a}=\hat{i}-\hat{j}, \bar{b}=\hat{j}-\hat{k}, \bar{c}=\hat{k}-\hat{i}$ then a unit vector $\bar{d}$ such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=0=[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{d}}]$ is

If $\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $|\overline{\mathrm{a}}+\overline{\mathrm{b}}|^2+|\overline{\mathrm{a}}+\overline{\mathrm{c}}|^2=8$, then $|\overline{\mathrm{a}}+3 \overline{\mathrm{~b}}|^2+|\overline{\mathrm...

The area of a parallelogram whose diagonals are the vectors $2 \bar{a}-\bar{b}$ and $4 \bar{a}-5 \bar{b}$, where $\bar{a}$ and $\bar{b}$ are unit vectors forming an angle of $45^{\circ}$ is

The projection of the line segment joining $\mathrm{P}(2,-1,0)$ and $\mathrm{Q}(3,2,-1)$ on the line whose direction ratios are $1,2,2$ is

If the lengths of three vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are $5,12,13$ units respectively, and each one is perpendicular to the sum of the other two, then $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=\ldots \...

If the area of parallelogram, whose diagonals are $\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\alpha \hat{\mathrm{k}}$ is $\frac{\sqrt{93}}{2}$ sq. units, then $\alpha=$

If $\bar{p}=2 \hat{i}+\hat{k}, \bar{q}=\hat{i}+\hat{j}+\hat{k}, \bar{r}=4 \hat{i}-3 \hat{j}+7 \hat{k}$ and a vector $\overline{\mathrm{m}}$ is such that $\overline{\mathrm{m}} \times \overline{\mathrm{q}}=\overline{\mathrm{r}} \times \overl...

The number of integral values of $p$ for which the vectors $(p+1) \hat{i}-3 \hat{j}+p \hat{k}, p \hat{i}+(p+1) \hat{j}-3 \hat{k}$ and $-3 \hat{\mathrm{i}}+\mathrm{p} \hat{\mathrm{j}}+(\mathrm{p}+1) \hat{\mathrm{k}}$ are linearly dependent v...

Let $\bar{a}, \bar{b}$, and $\bar{c}$ be unit vectors. Suppose that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0$ and if the angle between $\overline{\mathrm{b}}$ and $\overline{\mat...

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors such that $|\overline{\mathrm{a}}+\overline{\mathrm{b}}|=\sqrt{3}$, then the angle between $\bar{a}$ and $\bar{b}$ is

The line of intersection of the planes $\bar{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1 \quad$ and $\quad \bar{r} \cdot(\hat{i}+4 \hat{j}-2 \hat{k})=2 \quad$ is parallel to the vector

The lines $\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$ and $\overline{\mathrm{r}}=\overline{\mathrm{c}}+\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$ will intersect...

Let $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ be unit vectors at an angle $\frac{\pi}{3}$ with each other. If $(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \...

The maximum value and minimum value of the volume of the parallelopiped having coterminous edges $\hat{\mathrm{i}}+x \hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{j}}+x \hat{\mathrm{k}}$ and $x \hat{\mathrm{i}}+\hat{\mathrm{k}}$ are respe...

Three vectors $\hat{\mathrm{i}}-\hat{\mathrm{k}}, \lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+(1-\lambda) \hat{\mathrm{k}}$ and $\mu \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+(1+\lambda-\mu) \hat{\mathrm{k}}$ represents coterminous edges of a...

The volume of the tetrahedron whose coterminous edges are represented by
\(\bar{a}=-12 \hat{i}+p \hat{k}, \bar{b}=3 \hat{j},-\hat{k}, \bar{c}=2 \hat{i}+\hat{j}-15 \hat{k}\)
570 cu. units, then $\mathrm{p}=$

The value of $m \in \mathbb{R}$, when angle between the vectors $\overline{\mathrm{p}}=\mathrm{m} y \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overline{\mathrm{q}}=y \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{~m} y \ha...

$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are nonzero vectors such that $\overline{\mathrm{a}}$ is perpendicular to $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}},|\overline{\mathrm{a}}|=1,|\overline{\mathrm{...

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\overline{\mathrm{a}}+\overline{\mathrm{b}}$ is a unit vector when $\theta$ is

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=4, \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=2$. If $\overline{\mathrm{c}}=(2 \overline{\mathrm{a}} \times \overline{\mathrm{b}...

The magnitude of a vector which is orthogonal to the vector $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and is coplanar with the vectors $\hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}+\hat{k}$ is

If $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are unit vectors such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\frac{1}{2}, \overline{\mathrm{c}} \cdot \overline{\mathrm{d}}=\frac{1}{2}$ and the angle between $\overline{\mathrm{a}} \...

If $\bar{a}=4 \hat{i}+3 \hat{j}+\hat{k}, \bar{b}=\hat{i}-2 \hat{j}+2 \hat{k}$ then $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{b}})))=$

If $\bar{a}=\hat{i}+\hat{j}, \bar{b}=2 \hat{i}-\hat{k}$ then the point of intersection of the lines $\overline{\mathrm{r}} \times \overline{\mathrm{a}}=\overline{\mathrm{b}} \times \overline{\mathrm{a}}$ and $\overline{\mathrm{r}} \times \o...

If the points $\mathrm{A}(1,1,2), \mathrm{B}(2,1, \mathrm{p}), \mathrm{C}(1,0,3)$ and $D(2,2,0)$ are coplanar then the value of $p$ is

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\tan \frac{\theta}{2}=$

If $\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\bar{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$ then the value of $(2 \overline{\mathrm{a}}-\overline{\mathrm{b}}) \cdot[(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \time...

If the projection of $\bar{a}$ on $\bar{b}+\bar{c}$ is twice the projection of $\bar{b}+\bar{c}$ on $\bar{a}$ also if $|\bar{b}|=2 \sqrt{2},|\bar{c}|=4$ and the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{\pi...

The volume of the tetrahedron whose co-terminus edges are $\bar{a}, \bar{b}, \bar{c}$ is 12 cubic units. If the scalar projection of $\bar{a}$ on $\bar{b} \times \bar{c}$ is 4 , then $|\overline{\mathrm{b}} \times \overline{\mathrm{c}}|=$

If $\left[\begin{array}{lll}2 \bar{p}-3 \bar{r} & \bar{q} & \bar{s}\end{array}\right]+\left[\begin{array}{lll}3 \bar{p}+2 \bar{q} & \bar{r} & \bar{s}\end{array}\right]=m\left[\begin{array}{lll}\bar{p} & \bar{r} & \bar{s}\end{array}\right] +...

The projection of the line segment joining the points $(2,1,-3)$ and $(-1,0,2)$ on the line whose direction ratios are $3,2,6$ is

Let $\overline{\mathrm{OA}}=\overline{\mathrm{a}}, \overline{\mathrm{OB}}=\overline{\mathrm{b}}$ and if the vector along the angle bisector of $\angle \mathrm{AOB}$ is given by $x \frac{\overline{\mathrm{a}}}{|\overline{\mathrm{a}}|}+y \fra...

Let $\bar{u}, \bar{v}, \bar{w}$ be the vectors such that $|\overline{\mathrm{u}}|=1,|\overline{\mathrm{v}}|=2,|\overline{\mathrm{w}}|=3$. If the projection $\overline{\mathrm{v}}$ along $\overline{\mathrm{u}}$ is equal to that of $\overline...

In triangle ABC , the point P divides BC internally in the ratio $3: 4$ and Q divides CA internally in the ratio $5: 3$. If AP and BQ intersect in a point $G$, then $G$ divides $A P$ internally in the ratio

If $\bar{a}, \bar{b}, \bar{c}$ are three vectors such that $|\bar{a}|=3$, $|\bar{b}|=5,|\bar{c}|=7$ then $|\bar{a}-\bar{b}|^2+|\bar{b}-\bar{c}|^2+|\bar{c}-\bar{a}|^2$ does not exceed

If the area of the parallelogram with $\bar{a}$ and $\bar{b}$ as two adjacent sides is 15 square units, then the area (in square units) of the parallelogram, having $3 \bar{a}+2 \bar{b}$ and $\bar{a}+3 \bar{b}$ as two adjacent sides, is

If $\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=1$ and $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$, then $\overline{\...

Let $\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$
If $\bar{c}$ is a vector such that $\bar{a} \cdot \bar{c}=|\bar{c}|$,
$|\overline{\mathrm{c}}-\...

Suppose that $\bar{p}, \bar{q}$ and $\overline{\mathrm{r}}$ are three non-coplanar vectors in $\mathbb{R}^3$. Let the components of a vector $\overline{\mathrm{s}}$ along $\overline{\mathrm{p}}, \overline{\mathrm{q}}$ and $\overline{\mathrm...

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are three non-coplanar vectors, then $(\bar{a}+\bar{b}+\bar{c}) \cdot[(\bar{a}+\bar{b}) \times(\bar{a}+\bar{c})]$ equals

If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}, \hat{i}+\hat{j}+c \hat{k}$ $(a \neq b, c \neq 1)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ has the value __________.

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be vectors of magnitude 2,3 and 4 respectively. If $\bar{a}$ is perpendicular to $(\bar{b}+\bar{c}), \bar{b}$ is perpendicular to $(\bar{c}+\bar{a})$ and $\bar{c...