Vector Algebra

Let ABCD be a quadrilateral with $\overline{AB} = \vec{a}$, $\overline{AD} = \vec{b}$ and $\overline{AC} = 3\vec{a} + 2\vec{b}$. If its area is $\alpha$ times the area of the parallelogram with AB, AD as adjacent sides, then the value of $\...

The altitude of the parallelopiped, whose coterminous edges are the vectors $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - \hat{k}$, $\vec{c} = \hat{i} + \hat{j} + 3\hat{k}$, where $\vec{a}$, $\vec{b}$ are the sid...

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = |\vec{b}| = |\vec{c}| = 3$ and $\theta$ is the angle between $\vec{b}$ and $\vec{c}$ then $\tan^2\theta + \cot^2\theta =$

A unit vector coplanar with $\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} + \hat{k}$ and perpendicular to $\hat{i} + \hat{j} + \hat{k}$ is

A vector $\bar{r}$ of magnitude $3\sqrt{2}$ units which makes angles of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{2}$ respectively with Y and Z axes is

If a parallelogram is constructed on the vectors $\bar{a} = 3\bar{p} - \bar{q}, \bar{b} = \bar{p} + 3\bar{q}$ and $|\bar{p}| = 3, |\bar{q}| = 2$ and angle between $\bar{p}$ and $\bar{q}$ is $\dfrac{\pi}{3}$, then the ratio of the lengths of...

Let $\bar{a}, \bar{b}, \bar{c}$ be unit vectors such that $\bar{a}$ is perpendicular to the plane of $\bar{b}$ and $\bar{c}$. If the angle between $\bar{b}$ and $\bar{c}$ is $\dfrac{\pi}{3}$, then $|\bar{a} + \bar{b} + \bar{c}| =$

The volume of a tetrahedron with vertices $5\hat{i} - \hat{j} + \hat{k}, 7\hat{i} - 4\hat{j} + p\hat{k}, \hat{i} - 6\hat{j} + 10\hat{k}$ and $-\hat{i} - 3\hat{j} + 7\hat{k}$ is 11 cubic units, then one of the values of p is

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors having magnitudes 1, 1 and 2 respectively. If $\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}$, then the acute angle between $\bar{a}$ and $\bar{c}$ is

If $\bar{a}, \bar{b}, \bar{c}$ are three non zero and non-coplanar vectors such that $\bar{a} \times (\bar{b} \times \bar{c}) = \dfrac{\bar{b}}{2}$, then the angle between $\bar{a}$ and $\bar{b}$ is ...

If $\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar unit vectors such that the angle between any two of them is $60^\circ$, and the vector $\bar{d} = x\bar{a} + y\bar{b} + z\bar{c}$ is perpendicular to both $\bar{a}$ and $\bar{b}$, then th...

If $\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i}, \bar{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ with $c_1 = 1$, $c_2 = 2$, then value of $c_3$ such that $\bar{a}, \bar{b}, \bar{c}$ are coplanar is ____

If $\bar{a}$ and $\bar{b}$ have the same magnitude and angle between them is $60^\circ$ and their scalar product is $\dfrac{1}{2}$, then $|\bar{a}|$ is ____

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

If D and E are the midpoints of the sides BA and BC of triangle ABC, then $\overline{AE} + \overline{DC} = $

Let $A(2,3,0)$, $B(0,3,2)$ and $C(4,0,3)$ be vertices of a triangle, then the area of the triangle is

If $a, b, c$ are distinct non-negative numbers and the vectors $a\hat{i} + a\hat{j} + c\hat{k}, \hat{i} + \hat{k}, c\hat{i} + c\hat{j} + b\hat{k}$ lie in the same plane, then the value of $c$ is...

If $7\hat{j} + 10\hat{k}, -\hat{i} + 6\hat{j} + 6\hat{k}$ and $-4\hat{i} + 9\hat{j} + 6\hat{k}$ are the position vectors of the vertices A, B and C repectively of $\triangle ABC$. Then the position vector of the point where the bisector of ...

The vector $\bar{a} + 3\bar{b}$ is perpendicular to $7\bar{a} - 5\bar{b}$ and the vector $\bar{a} - 4\bar{b}$ is perpendicular to $7\bar{a} - 2\bar{b}$. Then the angle between $\bar{a}$ and $\bar{b}$ is

The parallelopiped is determined by vectors $\bar{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}$, $\bar{b} = \hat{i} + 3\hat{j} - 2\hat{k}$, $\bar{c} = -3\hat{i} + \hat{j} + 4\hat{k}$. The altitude of parallelopiped on the parallelogram base determi...

Let $x_0$ be the point of local maxima of $f(x) = \bar{a} \cdot (\bar{b} \times \bar{c})$ where $\bar{a} = x\hat{i} - 2\hat{j} + 3\hat{k}, \bar{b} = -2\hat{i} + x\hat{j} - \hat{k}$ and $\bar{c} = 7\hat{i} - 2\hat{j} + x\hat{k}$ then the val...

Let $\bar{a}, \bar{b}, \bar{c}$ be the unit vectors such that $\bar{a}$ is perpendicular to $\bar{b}$ and the angle between $\bar{b}$ and $\bar{c}$ is $120^\circ$. If $\bar{a} + \bar{c}$ is perpendicular to $\bar{b} + \bar{c}$ then

If $|\bar{a}| = 4, |\bar{b}| = 3$ and $\bar{a} \cdot \bar{b} = 8$ then $[\bar{a}\ \ \bar{a} + \bar{b}\ \ \bar{a} \times \bar{b}] = $

If $\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i} - \hat{j} + 2\hat{k}$ and $\bar{c} = x\hat{i} + (x - 2)\hat{j} - \hat{k}$ are three vectors in which $\bar{c}$ lies in the plane of $\bar{a}$ and $\bar{b}$, then $x = \cdots$

ABCD is a quadrilateral with $\overline{\mathrm{AB}}=\overline{\mathrm{a}}, \overline{\mathrm{AD}}=\overline{\mathrm{b}}$ and $\overline{\mathrm{AC}}=2 \overline{\mathrm{a}}+3 \overline{\mathrm{~b}}$. If its area is $\alpha$ times the area ...

Two adjacent sides of a parallelogram $A B C D$ are given by $\overline{\mathrm{AB}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}$ and $\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$. The side ...

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

If $\overline{\mathrm{c}}=5 \overline{\mathrm{a}}+6 \overline{\mathrm{~b}}$ and $3 \overline{\mathrm{c}}=\overline{\mathrm{a}}-4 \overline{\mathrm{~b}}$ then

If $\quad \overline{\mathrm{a}}=\lambda x \hat{\mathrm{i}}+y \hat{\mathrm{j}}+4 z \hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=y \hat{\mathrm{i}}+x \hat{\mathrm{j}}+3 y \hat{\mathrm{k}}$, $\overline{\mathrm{c}}=-z \hat{\mathrm{i}}-2 z \hat...

The volume of tetrahedron with co-terminus edges $\bar{a}, \bar{b}, \bar{c}$ is $\frac{64}{3}$ cubic units, then volume of parallelopiped considering co-terminus edges given by the vectors $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$...

Let $\quad \bar{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=3 \hat{i}-\hat{j}+\beta \hat{k} \quad$ and $\bar{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in \mathbb{R}$, be three vectors. If the projection of $\bar{a}$ on $\bar{c}...

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 $(\overline{\mathrm{b}}+\overline{\mathrm{c}}), \overline{\mathrm{b}}$ is perpen...

If $\bar{a}, \bar{b}, \bar{c}$ are three coplanar vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}=8$, the angle between $\overline{\mathrm{b}}$ and $\overline{\math...

In the above figure, P divides AC in the ratio $3: 4$ and Q divides BC in the ratio $4: 3$. Then M divides AQ in the ratio

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are three coplanar vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}=8$ and the angle between...

The unit vectors perpendicular to the plane determined by the points $\mathrm{A}(1,-1,2), \mathrm{B}(2,0,-1)$, $\mathrm{C}(0,2,1)$ is

The vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are such that $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{~b}}|=4,|\overline{\mathrm{c}}|=4$. If the projection of $\overline{\mathrm{b}}$ on $\overline{\mathrm{a}}$ is equal to projection of $...

The values of $x$ for which the angle between the vectors $\overline{\mathrm{a}}=2 x^2 \hat{\mathrm{i}}+4 x \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$ is obtuse, a...

The position vectors of the points $A, B, C$ are $\hat{i}+2 \hat{j}-\hat{k}, \hat{i}+\hat{j}+\hat{k}, 2 \hat{i}+3 \hat{j}+2 \hat{k}$ respectively. If $A$ is chosen as the origin, then the cross product of position vectors of $B$ and $C$ are

If the area of a parallelogram whose diagonals are represented by vectors $3 \hat{i}+\lambda \hat{j}+2 \hat{k}$ and $\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ is $\frac{\sqrt{117}}{2}$ sq. units, then $\lambda=$

If $\bar{a}, \bar{b}, \bar{c}$ are non coplanar unit vectors such that $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\frac{\overline{\mathrm{b}}+\overline{\mathrm{c}}}{\sqrt{2}}$ then the angle between $\...

A tetrahedron has vertices $\mathrm{O}(0,0,0), \mathrm{A}(1,2,1)$, $B(2,1,3), C(-1,1,2)$. Then the angle between the faces OAB and ABC will be

If $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{j}-\hat{k}$ then a vector $\bar{c}$ such that $\overline{\mathrm{a}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=3$ is

The altitude through vertex $A$ of $\triangle A B C$ with position vectors of points $A, B, C$ as $\bar{a}, \bar{b}, \bar{c}$ respectively is

If the vectors $\overline{\mathrm{a}}=\mathrm{c}\left(\log _7 x\right) \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \quad$ and $\overline{\mathrm{b}}=\left(\log _\gamma x\right) \hat{\mathrm{i}}+3 \mathrm{c}\left(\log _\gamma x\ri...

Let $\bar{a}=\hat{i}+\hat{j}-\hat{k}$ and $\bar{c}=5 \hat{i}-3 \hat{j}+2 \hat{k}$ and if $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=\overline{\mathrm{a}}$ then $|\overline{\mathrm{b}}|=$

If $\theta$ is an obtuse angle between vectors $\bar{a}$ and $\overline{\mathrm{b}}$ such that $|\overline{\mathrm{a}}|=5,|\overline{\mathrm{~b}}|=3$ and $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=5 \sqrt{5}$ then $\bar{a} \cdot ...

If $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are unit vectors and $|\overline{\mathrm{a}}|=7$, $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}} \times(\overline{\mathrm{c}} \t...

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}, \overline{\mathrm{d}}$ are vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overlin...

If $\bar{a}, \bar{b}, \bar{c}$ are three vectors such that $|\overrightarrow{\mathrm{a}}|=\sqrt{31}, 4|\overrightarrow{\mathrm{~b}}|=|\overrightarrow{\mathrm{c}}|=2$ and $2(\overline{\mathrm{a}} \times \overline{\mathrm{b}})=3(\overline{\ma...