Vector Algebra

Consider the vectors

$$\vec x = \hat i + 2\hat j + 3\hat k,\qquad \vec y = 2\hat i + 3\hat j + \hat k,\qquad\text{and}\qquad \vec z = 3\hat i + \hat j + 2\hat k.$$

For two distinct positive real numbers $\alpha$ and $\beta$, define

$$\vec X = \alpha\vec x + \beta\vec y - \vec z,\qquad \vec Y = \alpha\vec y + \beta\vec z - \vec x,\qquad\text{and}\qquad \vec Z = \alpha\vec z + \beta\vec x - \vec y.$$

If the vectors $\vec X$, $\vec Y$, and $\vec Z$ lie in a plane, then the value of $\alpha + \beta - 3$ is _____.

Let $\overrightarrow{OP} = \dfrac{\alpha-1}{\alpha}\hat{i} + \hat{j} + \hat{k}$, $\overrightarrow{OQ} = \hat{i} + \dfrac{\beta-1}{\beta}\hat{j} + \hat{k}$ and $\overrightarrow{OR} = \hat{i} + \hat{j} + \dfrac{1}{2}\hat{k}$ be three vectors, where $\alpha, \beta \in \mathbb{R} - \{0\}$ and $O$ denotes the origin. If $\left(\overrightarrow{OP} \times \overrightarrow{OQ}\right) \cdot \overrightarrow{OR} = 0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3x + 3y - z + l = 0$, then the value of $l$ is _____.

Let $\gamma \in \mathbb{R}$ be such that the lines $L_1 : \dfrac{x+11}{1} = \dfrac{y+21}{2} = \dfrac{z+29}{3}$ and $L_2 : \dfrac{x+16}{3} = \dfrac{y+11}{2} = \dfrac{z+4}{\gamma}$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O = (0, 0, 0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_1$ and $L_2$.

Match each entry in List-I to the correct entry in List-II.

The correct option is

Let $\vec p = 2\hat i + \hat j + 3\hat k$ and $\vec q = \hat i - \hat j + \hat k$. If for some real numbers $\alpha$, $\beta$, and $\gamma$, we have

$$15\hat i + 10\hat j + 6\hat k = \alpha\left(2\vec p + \vec q\right) + \beta\left(\vec p - 2\vec q\right) + \gamma\left(\vec p \times \vec q\right),$$

then the value of $\gamma$ is _____.

Let $Q$ be the cube with the set of vertices $\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \in \{0, 1\}\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_1$ and $\ell_2$, let $d(\ell_1, \ell_2)$ denote the shortest distance between them. Then the maximum value of $d(\ell_1, \ell_2)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is

Let $P$ be the plane $\sqrt3x + 2y + 3z = 16$ and let

$$S = \left\{\alpha\hat i + \beta\hat j + \gamma\hat k : \alpha^2 + \beta^2 + \gamma^2 = 1 \text{ and the distance of } (\alpha, \beta, \gamma) \text{ from the plane } P \text{ is } \frac72\right\}.$$

Let $\vec u$, $\vec v$ and $\vec w$ be three distinct vectors in $S$ such that $|\vec u - \vec v| = |\vec v - \vec w| = |\vec w - \vec u|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec u$, $\vec v$ and $\vec w$. Then the value of $\dfrac{80}{\sqrt3}V$ is _____.

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec a = \hat i + 2\hat j - 5\hat k$, $\vec b = 3\hat i + 6\hat j + 3\hat k$, $\vec c = \dfrac{17}{5}\hat i + \dfrac{16}{5}\hat j + 7\hat k$ and $\vec d = 2\hat i + \hat j + \hat k$, respectively. Then which of the following statements is true?

Let $\hat i, \hat j$ and $\hat k$ be the unit vectors along the three positive coordinate axes. Let
$$\vec a = 3\hat i + \hat j - \hat k,\qquad \vec b = \hat i + b_2\hat j + b_3\hat k,\ b_2, b_3 \in \mathbb{R},\qquad \vec c = c_1\hat i + c_2\hat j + c_3\hat k,\ c_1, c_2, c_3 \in \mathbb{R}$$
be three vectors such that $b_2b_3 > 0$, $\vec a\cdot\vec b = 0$ and
$$\begin{pmatrix}0 & -c_3 & c_2\\ c_3 & 0 & -c_1\\ -c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix}1\\ b_2\\ b_3\end{pmatrix} = \begin{pmatrix}3 - c_1\\ 1 - c_2\\ -1 - c_3\end{pmatrix}.$$
Then, which of the following is/are TRUE?

Let $\hat a,\hat b$ be unit vectors. If $\vec c$ be a vector such that the angle between $\hat a$ and $\vec c$ is $\dfrac\pi{12}$, and $\hat b=\vec c+2\left(\vec c\times\hat a\right)$, then $\left|6\vec c\right|^2$ is equal to :

If the shortest distance between the lines $\vec r=\left(-\hat i+3\hat k\right)+\lambda\left(\hat i-a\hat j\right)$ and $\vec r=\left(-\hat j+2\hat k\right)+\mu\left(\hat i-\hat j+\hat k\right)$ is $\sqrt{\dfrac23}$, then the integral value of $a$ is equal to ______.

If the shortest distance between the lines $\dfrac{x-1}2=\dfrac{y-2}3=\dfrac{z-3}\lambda$ and $\dfrac{x-2}1=\dfrac{y-4}4=\dfrac{z-5}5$ is $\dfrac1{\sqrt3}$, then the sum of all possible values of $\lambda$ is:

Let $\hat a$ and $\hat b$ be two unit vectors such that $\left|\left(\hat a+\hat b\right)+2\left(\hat a\times\hat b\right)\right|=2$. If $\theta\in(0,\pi)$ is the angle between $\hat a$ and $\hat b$, then among the statements:

(S1) : $2\left|\hat a\times\hat b\right|=\left|\hat a-\hat b\right|$

(S2) : The projection of $\hat a$ on $\left(\hat a+\hat b\right)$ is $\frac12$

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 $\theta$ be the angle between the vectors $\vec a$ and $\vec b$, where $|\vec a|=4$, $|\vec b|=3$ and $\theta\in\left(\dfrac\pi4,\dfrac\pi3\right)$. Then
$$\left|\left(\vec a-\vec b\right)\times\left(\vec a+\vec b\right)\right|^2+4\left(\vec a\cdot\vec b\right)^2$$
is equal to ______.

Let the lines
$$L_1:\vec r=\lambda\left(\hat i+2\hat j+3\hat k\right),\ \lambda\in\mathbb R$$
$$L_2:\vec r=\left(\hat i+3\hat j+\hat k\right)+\mu\left(\hat i+\hat j+5\hat k\right);\ \mu\in\mathbb R,$$
intersect at the point S. If a plane $ax+by-z+d=0$ passes through S and is parallel to both the lines $L_1$ and $L_2$, then the value of $a+b+d$ is equal to ______.

Let $\vec b=\hat i+\hat j+\lambda\hat k$, $\lambda\in\mathbb R$. If $\vec a$ is a vector such that $\vec a\times\vec b=13\hat i-\hat j-4\hat k$ and $\vec a\cdot\vec b+21=0$, then $\left(\vec b-\vec a\right)\cdot\left(\hat k-\hat j\right)+\left(\vec b+\vec a\right)\cdot\left(\hat i-\hat k\right)$ is equal to ______.

If $\vec a\cdot\vec b=1$, $\vec b\cdot\vec c=2$ and $\vec c\cdot\vec a=3$, then the value of $\left[\vec a\times\left(\vec b\times\vec c\right),\ \vec b\times\left(\vec c\times\vec a\right),\ \vec c\times\left(\vec b\times\vec a\right)\right]$ is :

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 $\vec a=\hat i+\hat j-\hat k$ and $\vec c=2\hat i-3\hat j+2\hat k$. Then the number of vectors $\vec b$ such that $\vec b\times\vec c=\vec a$ and $\left|\vec b\right|\in\{1,2,\dots,10\}$ is :

The shortest distance between the lines $\dfrac{x-3}2=\dfrac{y-2}3=\dfrac{z-1}{-1}$ and $\dfrac{x+3}2=\dfrac{y-6}1=\dfrac{z-5}3$, is