Vector operations

Let the vectors $\vec a=-\hat i+\hat j+3\hat k$ and $\vec b=\hat i+3\hat j+\hat k$. For some $\lambda,\mu\in\mathbb R$, let $\vec c=\lambda\vec a+\mu\vec b$. If $\vec c\cdot\left(3\hat i-6\hat j+2\hat k\right)=10$ and $\vec c\cdot\left(\hat i+\hat j+\hat k\right)=-2$, then $|\vec c|^2$ is equal to :

Let $\vec{a_k}=(\tan\theta_k)\hat i+\hat j$ and $\vec{b_k}=\hat i-(\cot\theta_k)\hat j$, where $\theta_k=\dfrac{2^{k-1}\pi}{2^n+1}$, for some $n\in\mathbb N$, $n>5$. Then the value of $\dfrac{\sum_{k=1}^n\left|\vec{a_k}\right|^2}{\sum_{k=1}^n\left|\vec{b_k}\right|^2}$ is ________.

Let O be the origin, $\overrightarrow{OP}=\vec a$ and $\overrightarrow{OQ}=\vec b$. If R is the point on $\overrightarrow{OP}$ such that $\overrightarrow{OP}=5\overrightarrow{OR}$, and M is the point such that $\overrightarrow{OQ}=5\overrightarrow{RM}$, then $\overrightarrow{PM}$ is equal to :

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec 0$ denote the zero vector. Let $P$, $Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\Delta PQR$ such that

$$\overrightarrow{SP} + 5\,\overrightarrow{SQ} + 6\,\overrightarrow{SR} = \vec 0.$$

Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of

$$\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$$

is _____.

Let $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $a_i>0$, $i=1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\vec a$ on the vector $3\hat i+4\hat j$ be 7. Let $\vec b$ be a vector obtained by rotating $\vec a$ with 90°. If $\vec a$, $\vec b$ and $x$-axis are coplanar, then projection of a vector $\vec b$ on $3\hat i+4\hat j$ is equal to :

Let $\vec a=\hat i+\hat j+2\hat k$, $\vec b=2\hat i-3\hat j+\hat k$ and $\vec c=\hat i-\hat j+\hat k$ be three given vectors. Let $\vec v$ be a vector in the plane of $\vec a$ and $\vec b$ whose projection on $\vec c$ is $\dfrac2{\sqrt3}$. If $\vec v\cdot\hat j=7$, then $\vec v\cdot\left(\hat i+\hat k\right)$ is equal to :

Let A, B, C be three points whose position vectors respectively are
$$\vec a=\hat i+4\hat j+3\hat k$$
$$\vec b=2\hat i+\alpha\hat j+4\hat k,\ \alpha\in\mathbb R$$
$$\vec c=3\hat i-2\hat j+5\hat k$$
If $\alpha$ is the smallest positive integer for which $\vec a$, $\vec b$, $\vec c$ are noncollinear, then the length of the median, in $\Delta ABC$, through A is :

Three lines

$$L_1:\ \vec r = \lambda\hat i,\ \lambda\in\mathbb R,$$

$$L_2:\ \vec r = \hat k + \mu\hat j,\ \mu\in\mathbb R\quad\text{and}$$

$$L_3:\ \vec r = \hat i + \hat j + v\hat k,\ v\in\mathbb R$$

are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P$, $Q$ and $R$ are collinear?

Suppose that $\vec p$, $\vec q$ and $\vec r$ are three non-coplanar vectors in $\mathbb R^3$. Let the components of a vector $\vec s$ along $\vec p$, $\vec q$ and $\vec r$ be 4, 3 and 5, respectively. If the components of this vector $\vec s$ along $(-\vec p+\vec q+\vec r)$, $(\vec p-\vec q+\vec r)$ and $(-\vec p-\vec q+\vec r)$ are $x$, $y$ and $z$, respectively, then the value of $2x+y+z$ is

Let $\overrightarrow{PR}=3\hat i+\hat j-2\hat k$ and $\overrightarrow{SQ}=\hat i-3\hat j-4\hat k$ determine diagonals of a parallelogram $PQRS$ and $\overrightarrow{PT}=\hat i+2\hat j+3\hat k$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{PT}$, $\overrightarrow{PQ}$ and $\overrightarrow{PS}$ is

Let $\vec a=\hat i+\hat j+\hat k$, $\vec b=\hat i-\hat j+\hat k$ and $\vec c=\hat i-\hat j-\hat k$ be three vectors. A vector $\vec v$ in the plane of $\vec a$ and $\vec b$, whose projection on $\vec c$ is $\dfrac1{\sqrt3}$, is given by

The vector(s) which is/are coplanar with vectors $\hat i+\hat j+2\hat k$ and $\hat i+2\hat j+\hat k$, and perpendicular to the vector $\hat i+\hat j+\hat k$ is/are

Let P, Q, R and S be the points on the plane with position vectors $-2\hat i-\hat j$, $4\hat i$, $3\hat i+3\hat j$ and $-3\hat i+2\hat j$ respectively. The quadrilateral PQRS must be a