Vectors Questions - JEE

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ have magnitudes 1, 2, 3 respectively, satisfying $|(\vec{a}\times\vec{b})\cdot\vec{c}|=6$. If $\vec{d}$ is a unit vector coplanar with $\vec{b}$ and $\vec{c}$ such that $\vec{b}\cdot\vec{d}=1$, then $|(\vec{a}\times\vec{c})\cdot\vec{d}|^2+|(\vec{a}\times\vec{c})\times\vec{d}|^2$ is

Let $ABC$ be a triangle and $\vec{a},\vec{b},\vec{c}$ be the position vectors of the point $A, B, C$ respectively. External bisectors of $\angle B$ and $\angle C$ meet at $P$ with the sides of the triangle as $a, b, c$, the position vector of $P$ becomes :

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar, mutually perpendicular unit vectors, then $[\vec{a} \vec{q}] \vec{a} + [\vec{b} \vec{q}] \vec{b} + [\vec{c} \vec{q}] \vec{c}$ is equal to:

If vectors $\vec{a} = \frac{\vec{i} + \vec{j}}{\sqrt{2}}, \vec{b} = \frac{-\vec{i} + \vec{j}}{\sqrt{2}}$ and $\vec{c} = \vec{k}$ then the value of $(\vec{r} \cdot \vec{a})^2 + (\vec{r} \cdot \vec{b})^2 + (\vec{r} \cdot \vec{c})^2$ is equal to:

If $\vec{a}$ is a unit vector and projection of $\vec{x}$ along $\vec{a}$ is 2 units and $(\vec{a} \times \vec{x}) \cdot \vec{b} = \vec{x}$, then $\vec{x}$ is given by:

A vector $\vec{a} = a\vec{i} + b\vec{j} + c\vec{k}$ is said to be rational vector. If $a, b, c$ are all rational. If a rational vector with magnitude as positive integer makes an angle $\pi/4$ with vector $\vec{b} = \sqrt{2}\vec{i} + \sqrt{2}\vec{j} + \vec{k}$, then $\vec{a}$:

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-coplanar vectors and $\vec{d}$ be a non-zero vector, which is perpendicular to $(\vec{a} + \vec{b} + \vec{c})$. Now if $\vec{d} = \sin x(\vec{a} \times \vec{b}) + \cos y(\vec{b} \times \vec{c}) + 2(\vec{c} \times \vec{a})$, then minimum value of $x^2 + y^2$ is equal to:

The position vectors of the vertices $A, B$ and $C$ of a triangle are three unit vectors $\vec{a}, \vec{b}$ and $\vec{c}$. A vector $\vec{d}$ is such that $\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c}$ and $\vec{d} = \lambda(\vec{b} + \vec{c})$, then triangle $ABC$ is:

If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are four non-coplanar unit vectors, $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular, such that $\vec{d}$ makes equal angles with all the three vectors $\vec{a}, \vec{b}, \vec{c}$, then:

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ be such that $\vec{c} - 3\vec{a} = -\vec{b} - \vec{a}$. Then the points with position vectors as $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are:

Let $\vec{a} = a\vec{i} + b\vec{j} + c\vec{k}$ and $\vec{\beta} = b\vec{i} + \vec{c}\vec{j} + a\vec{k}$, where $a, b, c \in \mathbb{R}$. If '$\theta$' be the angle between $\vec{a}$ and $\vec{\beta}$ then:

If the lines $\vec{r} = \vec{a} + (\vec{b} \times \vec{c})$, and $\vec{r} = \vec{b} + s(\vec{c} \times \vec{a})$ intersect (t and s are scalars) then:

If $4\vec{a} + 5\vec{b} + 9\vec{c} = \vec{0}$, then $(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})$ is equal to:

Given that $\vec{a}$ is perpendicular to $\vec{b}$ and p is a non-zero scalar, if $p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c}$ then $\vec{r} = $:

If the plane faces of a tetrahedon are represented by the equations $\vec{r} \cdot (\vec{i} + \vec{j}) = 0$, $\vec{r} \cdot (n\vec{k} + m\vec{j}) = 0$, $\vec{r} \cdot (m\vec{k} + \vec{i}) = 0$ and $\vec{r} \cdot (\vec{i} + m\vec{j} + n\vec{k}) = p$, then the volume of the tetrahedon is:

Let $\vec{a} = \vec{i} + \vec{j}$ and $\vec{b} = 2\vec{i} - \vec{k}$. The point of intersection of the lines $\vec{r} \times \vec{a} = \vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$ is:

If the non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendiculars to each other then the solution of the equation $\vec{r} \times \vec{a} = \vec{b}$ is given by:

Let $OPQR$ is a tetrahedon such that O is origin and $\vec{p}, \vec{q}, \vec{r}$ are position vectors of P, Q, R respectively and $\alpha$ is the angle which OP makes with face PQR then:

If $\vec{r} = l(\vec{b} \times \vec{c}) + m(\vec{c} \times \vec{a}) + n(\vec{a} \times \vec{b})$ and $[\vec{b}\vec{c}\vec{c}] = 2$, then $l + m + n$ is equal to:

If vectors $\vec{b} = (\tan\alpha, -1, 2\sqrt{\sin\frac{\alpha}{2}})$ and $\vec{c} = (\tan\alpha, \tan\alpha, -\frac{3}{\sqrt{\sin\alpha/2}})$ are orthogonal and vector $\vec{a} = (1, 3, \sin2\alpha)$ makes an obtuse angle with the z-axis then:

Let position vector of point $A$ be $\vec{i} + \vec{j} + \vec{k}$ and that of point $B$ be $-\vec{i} + \vec{k}$, then the position vector of point $R(\vec{r})$ such that $AR$ is perpendicular to $BR$ and $\vec{r}$ is not perpendicular to $\vec{r} - (\vec{j} + 2\vec{k})$ is:

If three coterminous edges of a tetrahedron are $\vec{a}, \vec{b}, \vec{c}$ such that $|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 4$, angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$, $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{4}$ and $\vec{c}$ and $\vec{a}$ is $\frac{\pi}{6}$. The area of the base is $2$ sq. units, then the height of the tetrahedron is:

If $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \cdot (\vec{c} \times \vec{b}) = 0$ then which of the following is always true:

If $\vec{\alpha}$ and $\vec{\beta}$ be two perpendicular unit vectors such that $\vec{x} = \vec{\beta} - (\vec{a} \times \vec{x})$, then $|\vec{x}|$ is equal to:

If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are on a circle of radius $R$ whose centre is at origin and $\vec{c} - \vec{a}$ is perpendicular to $\vec{d} - \vec{b}$, then $|\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2$ (AC is diameter)

If $ABC$ be a triangle of sides $a, b, c$ with position vectors of $A, B, C$ as $\vec{a}, \vec{b}$ and $\vec{c}$ respectively, then the position vector of its incentre is:

The vector sum of $\vec{a}$ and $\vec{b}$ trisects the angle $\theta$ between them. If $|\vec{a}| = a; |\vec{b}| = b; a > b$, then:

A line passes through the points whose position vectors are $\vec{r} + \vec{j} - 2\vec{k}$ and $\vec{r} - 3\vec{j} + \vec{k}$. The position vector of a point on it at a unit distance from the first point is:

The volume of a right triangular prism $ABC A_1 B_1 C_1$ is equal to 3. If the position vectors of the vertices of the base $ABC$ are $A(1, 0, 1); B(2,0, 0)$ and $C(0, 1, 0)$ the position vectors of the vertex $A_1$ can be:

If non-zero vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ satisfy $(\vec{a} \times \vec{b}) \cdot \vec{c} = |\vec{a}||\vec{b}||\vec{c}|$ holds then:

Three points having position vectors $\vec{a}, \vec{b}$ and $\vec{c}$ will be collinear if:

If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are any four vectors, then $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})$ is a vector:

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar, mutually perpendicular unit vectors, then $[\vec{a} \vec{q}] \vec{a} + [\vec{b} \vec{q}] \vec{b} + [\vec{c} \vec{q}] \vec{c}$ is equal to:

If vectors $\vec{a} = \frac{\vec{i} + \vec{j}}{\sqrt{2}}, \vec{b} = \frac{-\vec{i} + \vec{j}}{\sqrt{2}}$ and $\vec{c} = \vec{k}$ then the value of $(\vec{r} \cdot \vec{a})^2 + (\vec{r} \cdot \vec{b})^2 + (\vec{r} \cdot \vec{c})^2$ is equal to:

If $\vec{a}$ is a unit vector and projection of $\vec{x}$ along $\vec{a}$ is 2 units and $(\vec{a} \times \vec{x}) \cdot \vec{b} = \vec{x}$, then $\vec{x}$ is given by:

A vector $\vec{a} = a\vec{i} + b\vec{j} + c\vec{k}$ is said to be rational vector. If $a, b, c$ are all rational. If a rational vector with magnitude as positive integer makes an angle $\pi/4$ with vector $\vec{b} = \sqrt{2}\vec{i} + \sqrt{2}\vec{j} + \vec{k}$, then $\vec{a}$:

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-coplanar vectors and $\vec{d}$ be a non-zero vector, which is perpendicular to $(\vec{a} + \vec{b} + \vec{c})$. Now if $\vec{d} = \sin x(\vec{a} \times \vec{b}) + \cos y(\vec{b} \times \vec{c}) + 2(\vec{c} \times \vec{a})$, then minimum value of $x^2 + y^2$ is equal to:

Let $ABCD$ be a tetrahedron in which position vectors of $A, B, C$ and $D$ are $\vec{i} + \vec{j} + \vec{k}, 2\vec{i} + 2\vec{j} + 2\vec{k}, 3\vec{i} + 2\vec{j} + \vec{k}$ and $2\vec{i} + 3\vec{j} + 2\vec{k}$. If $ABC$ be the base of tetrahedron then height of tetrahedron is: