Cross product

Let a vector $\vec a$ has magnitude 9. Let a vector $\vec b$ be such that for every $(x,y)\in\mathbf R\times\mathbf R-\{(0,0)\}$, the vector $\left(x\vec a+y\vec b\right)$ is perpendicular to the vector $\left(6y\vec a-18x\vec b\right)$. Then the value of $\left|\vec a\times\vec b\right|$ is equal to :

Let $\hat a$ and $\hat b$ be two unit vectors such that the angle between them is $\dfrac\pi4$. If θ is the angle between the vectors $(\hat a+\hat b)$ and $(\hat a+2\hat b+2(\hat a\times\hat b))$, then the value of 164 cos$^2$θ is equal to :

Let $\vec a,\vec b,\vec c$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $(\vec a\times\vec b)\cdot(\vec b\times\vec c)+(\vec b\times\vec c)\cdot(\vec c\times\vec a)+(\vec c\times\vec a)\cdot(\vec a\times\vec b)=168$, then $|\vec a|+|\vec b|+|\vec c|$ is equal to :

Let $\vec a$ and $\vec b$ be two vectors such that $|\vec a+\vec b|^2=|\vec a|^2+2|\vec b|^2$, $\vec a\cdot\vec b=3$ and $|\vec a\times\vec b|^2=75$. Then $|\vec a|^2$ is equal to _______.

The torque of a force $5\hat i+3\hat j-7\hat k$ about the origin is τ. If the force acts on a particle whose position vector is $2\hat i+2\hat j+\hat k$, then the value of τ will be

Let $O$ be the origin and $\overrightarrow{OA} = 2\hat i + 2\hat j + \hat k$, $\overrightarrow{OB} = \hat i - 2\hat j + 2\hat k$ and $\overrightarrow{OC} = \frac12\left(\overrightarrow{OB} - \lambda\overrightarrow{OA}\right)$ for some $\lambda > 0$. If $\left|\overrightarrow{OB}\times\overrightarrow{OC}\right| = \frac92$, then which of the following statements is (are) TRUE?

In a triangle $PQR$, let $\vec a = \overrightarrow{QR}$, $\vec b = \overrightarrow{RP}$ and $\vec c = \overrightarrow{PQ}$. If

$$|\vec a| = 3,\qquad |\vec b| = 4\qquad\text{and}\qquad \frac{\vec a\cdot(\vec c - \vec b)}{\vec c\cdot(\vec a - \vec b)} = \frac{|\vec a|}{|\vec a| + |\vec b|},$$

then the value of $|\vec a\times\vec b|^2$ is ______

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ} = a\hat i + b\hat j$ and $\overrightarrow{PS} = a\hat i - b\hat j$ are adjacent sides of a parallelogram $PQRS$. Let $\vec u$ and $\vec v$ be the projection vectors of $\vec w = \hat i + \hat j$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$, respectively. If $|\vec u| + |\vec v| = |\vec w|$ and if the area of the parallelogram $PQRS$ is 8, then which of the following statements is/are TRUE?

Let $L_1$ and $L_2$ denote the lines

$$\vec r = \hat i + \lambda(-\hat i + 2\hat j + 2\hat k),\ \lambda\in\mathbb R\quad\text{and}\quad\vec r = \mu(2\hat i - \hat j + 2\hat k),\ \mu\in\mathbb R$$

respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

Three lines are given by

$$\vec r = \lambda\hat i,\ \lambda\in\mathbb R;\quad\vec r = \mu(\hat i + \hat j),\ \mu\in\mathbb R\quad\text{and}\quad\vec r = v(\hat i + \hat j + \hat k),\ v\in\mathbb R.$$

Let the lines cut the plane $x + y + z = 1$ at the points $A$, $B$ and $C$ respectively. If the area of the triangle $ABC$ is $\Delta$ then the value of $(6\Delta)^2$ equals ______

Let $\vec a = 2\hat i + \hat j - \hat k$ and $\vec b = \hat i + 2\hat j + \hat k$ be two vectors. Consider a vector $\vec c = \alpha\vec a + \beta\vec b$, $\alpha, \beta\in\mathbb R$. If the projection of $\vec c$ on the vector $(\vec a + \vec b)$ is $3\sqrt2$, then the minimum value of $(\vec c - (\vec a\times\vec b))\cdot\vec c$ equals ______

Let $P_1: 2x + y - z = 3$ and $P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b} = 0$. For some $x, y \in \mathbb{R}$, let $\vec{c} = x\vec{a} + y\vec{b} + (\vec{a} \times \vec{b})$. If $|\vec{c}| = 2$ and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then the value of $8\cos^2\alpha$ is _____ .

Consider the cube in the first octant with sides $OP$, $OQ$ and $OR$ of length 1, along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0, 0, 0)$ is the origin. Let $S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT$. If $\vec{p} = \overrightarrow{SP}$, $\vec{q} = \overrightarrow{SQ}$, $\vec{r} = \overrightarrow{SR}$ and $\vec{t} = \overrightarrow{ST}$, then the value of $|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|$ is _____ .

The equation of the plane passing through the point $(1, 1, 1)$ and perpendicular to the planes $2x + y - 2z = 5$ and $3x - 6y - 2z = 7$, is

$|\overrightarrow{OX} \times \overrightarrow{OY}| =$

Let $\vec a=2\hat i+\hat j-2\hat k$ and $\vec b=\hat i+\hat j$. Let $\vec c$ be a vector such that $|\vec c-\vec a|=3$, $\left|(\vec a\times\vec b)\times\vec c\right|=3$ and the angle between $\vec c$ and $\vec a\times\vec b$ be 30°. Then $\vec a\cdot\vec c$ is equal to :

The line of intersection of the planes $\vec r\cdot(3\hat i-\hat j+\hat k)=1$ and $\vec r\cdot(\hat i+4\hat j-2\hat k)=2$, is :

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat i-6\hat j$ and $3\hat i+4\hat j-12\hat k$, is :

In a certain region static electric and magnetic fields exist. The magnetic field is given by $\vec B=B_0\left(\hat i+2\hat j-4\hat k\right)$. If a test charge moving with a velocity $\vec v=v_0\left(3\hat i-\hat j+2\hat k\right)$ experiences no force in that region, then the electric field in the region, in SI units, is :