Vector Algebra - Class 12

Let $O$ be a fixed origin. A moving point $O_t$ has a position vector relative to $O$ given by: $\vec{r}_0(t) = \left( \int_{-1}^t \frac{u^2 - 1}{u^4 + 1} du + C \right) \hat{i} + t \hat{j} + t^2 \hat{k}$ The vertices of a tetrahedron $A, B, C, D$ have position vectors relative to $O_t$ such that their sum is: $\sum_{V \in \{A,B,C,D\}} \vec{r}_{V/O_t} = 4 \left( \int_t^1 \frac{u^2 - 1}{u^4 + 1} du \right) \hat{i} + 4(1-t) \hat{j} + 4(1-t^2) \hat{k}$ The parameter $t$ is restricted to the maximal domain $S$ where the function $f(t) = \int_0^t \frac{x^2 - 1}{x^4 + 1} dx$ is decreasing. The constant $C$ is chosen such that the maximum value of the $\hat{i}$-component of $\vec{r}_0(t)$ for $t \in S$ is $\sqrt{2} \ln(\sqrt{2} + 1)$. If the absolute position vector of the centroid $G$ of the tetrahedron is $\vec{r}_G = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$, find the value of $\alpha + \beta + \gamma$.

Let the quadratic equations $x^2 - ax + 6 = 0$ and $x^2 - bx + 12 = 0$ have a common root $\alpha$. Let their other roots be $\beta$ and $\gamma$ respectively. It is given that $\beta$ and $\gamma$ are positive integers such that $\beta + \gamma = 6$.

Let $P$ be a point whose position vector with respect to the origin $O(0,0,0)$ is $\vec{p} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$. The origin is then shifted to the point $Q(2, 0, 2)$. Let $\vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}$ be the position vector of $P$ with respect to the new origin.

If a matrix $M$ is defined as $M = \begin{pmatrix} 2 & v_1 & v_2 \\ -v_1 & 2 & v_3 \\ -v_2 & -v_3 & 2 \end{pmatrix}$, then the value of $\det(\text{adj}(M))$ is:

Let $S = \{ z \in \mathbb{C} : |z - 2| \le 1 \}$. If $\lambda$ is a real parameter such that the maximum value of $|\lambda z + 3i|$ for $z \in S$ is $7$, then the sum of all possible values of $\lambda$ is

Let $\vec{a} = \sin 3\theta \hat{i} + \cos 3\theta \hat{j} + \sin \theta \hat{k}$ and $\vec{b} = \sin \theta \hat{i} + \cos \theta \hat{j} + \sin 3\theta \hat{k}$ be two vectors forming adjacent sides of a triangle, where $\theta \in \left(0, \frac{\pi}{4}\right)$. If the area of the triangle is $\frac{\sqrt{3}}{2\sqrt{2}}$, then a unit vector perpendicular to the plane of the triangle is:

Let $S = \{0, 1\}$. Three vectors $\vec{a}, \vec{b}, \vec{c}$ in $\mathbb{R}^3$ are generated by choosing their components independently and uniformly at random from $S$. What is the probability that the volume of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ is non-zero?

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors and $\vec{a}', \vec{b}', \vec{c}'$ be their reciprocal system of vectors. Let $\vec{u} = \vec{a}' + 2\vec{b}' + 3\vec{c}'$, $\vec{v} = 2\vec{a}' - \vec{b}' + \vec{c}'$, and $\vec{w} = 3\vec{a}' + \vec{b}' + 5\vec{c}'$. If the scalar triple product $[\vec{u} \; \vec{v} \; \vec{w}] = \frac{5}{27}$, find the value of $[\vec{a} \; \vec{b} \; \vec{c}] - \sqrt{[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]}$.

A conservative vector field is given by $\vec{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j}$. It is known that $M(x,y) = 2x e^{x^2 - y^2} + 3y$ and the scalar potential function $\phi(x,y)$ associated with $\vec{F}$ satisfies $\phi(0,y) = e^{-y^2}$. Find the unit vector in the direction of $\vec{F}$ at the point $(1, 1)$.

Let $\vec{a}, \vec{b}$, and $\vec{c}$ be three unit vectors in $\mathbb{R}^3$ such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$. Suppose $\vec{c}$ satisfies the equation $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) (\vec{b} \times \vec{c})$ Find the maximum possible value of the volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b}$, and $\vec{c}$.

Let $\vec{a}_r = \left( \frac{10 - r}{r(r+1)(r+2)} - 1 \right) \hat{i} + r\hat{j} + r^2\hat{k}$ and $\vec{b}_r = \hat{i}$. Let $M_r$ be the $3 \times 3$ matrix representing the linear transformation $T_r(\vec{x}) = \vec{x} + \vec{a}_r \times (\vec{x} \times \vec{b}_r)$ for any vector $\vec{x} \in \mathbb{R}^3$. If $N_r = \text{adj}(M_r)$, find the value of $\sum_{r=1}^{20} \sqrt[4]{\det(N_r)}$.

Let $P$ be the point that divides the line segment joining the points $A(1, -2, 3)$ and $B(4, 7, -6)$ internally in the ratio $1:2$. Let $c$ be the perpendicular distance of point $P$ from the $yz$-plane. Consider the function $f(x) = x^2 - 5x - 3|x - c|$ defined on the interval $[1, 5]$. If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(x)$ on the given interval respectively, then the value of $M + m$ is:

Let $f(x) = \begin{vmatrix} x^2 + |x-1| & x^3 + |x-2| & x^4 + |x-3| \\ x^2 - |x-1| & x^3 - |x-2| & x^4 - |x-3| \\ x^2 & x^3 & x^4 \end{vmatrix}$. Let $D$ be the set of all real numbers $x$ at which $f(x)$ is not differentiable. If $D$ is a non-empty set, let $S = \sum_{x \in D} x$. If $D$ is the empty set, let $S = \sum_{k=1}^{5} f'(k)$. Find the value of $S$.

Let $T_k(x) = (x^2+x) \log_2\left( \frac{(k+1)^2}{k(k+2)} \right)$ for $k \in \mathbb{N}$, and define $S(x) = \sum_{k=1}^{\infty} T_k(x)$. Consider the following vectors in $\mathbb{R}^3$: $\vec{v}_1 = [S(x)] \hat{i} + S(x) \hat{j} + \hat{k}$ $\vec{v}_2 = S(x) \hat{i} + [S(x)] \hat{j} + \hat{k}$ $\vec{v}_3 = \hat{i} + \hat{j} + \hat{k}$ where $[ \cdot ]$ denotes the greatest integer function. Find the number of values of $x \in [-2, 2]$ for which the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ are linearly dependent.

Consider two lines $L_1$ and $L_2$ in $\mathbb{R}^3$ having direction vectors $\vec{d_1} = (\sin \alpha + \sqrt{3}\cos \alpha)\hat{i} + (\cos \alpha - \sqrt{3}\sin \alpha)\hat{j} + 2\hat{k}$ and $\vec{d_2} = (\sin \beta)\hat{i} + (\cos \beta)\hat{j} - \hat{k}$ respectively, for some $\alpha, \beta \in \mathbb{R}$. If the lines $L_1$ and $L_2$ are parallel, then the scalar projection of $\vec{d_1}$ on $\vec{d_2}$ is:

Let $A$ be a $3 \times 3$ symmetric matrix and $B$ be a $3 \times 3$ skew-symmetric matrix. Define two matrices $P = A^2 B - B A^2$ and $Q = A B^2 - B^2 A$. Which of the following statements is ALWAYS true regarding the determinants of $P$ and $Q$?

Let the vectors $\vec{v}_1(x)$, $\vec{v}_2(x)$, and $\vec{v}_3(x)$ be defined for $x > 0$ as: $\vec{v}_1(x) = \left( 2 \int_{1/e}^x \frac{|\ln t|}{t} dt \right) \hat{i} + \left( 2 \int_1^x \frac{|\ln t|}{t} dt \right) \hat{j} + \hat{k}$ $\vec{v}_2(x) = ((\ln x)^2 - \ln x) \hat{i} + (\ln x + 2) \hat{j} + \hat{k}$ $\vec{v}_3(x) = \hat{i} + 2 \hat{j}$ Let $S$ be the set of all real numbers $x > 0$ for which the vectors $\vec{v}_1(x), \vec{v}_2(x), \vec{v}_3(x)$ are linearly dependent. Find the product of all elements in $S$.

Consider three points $A(2, -1, 4)$, $B(-2, 3, -3)$, and $C(-1, 1, -5)$ in 3D space. Let $p$ be the scalar projection of vector $\vec{BA}$ on vector $\vec{BC}$, and $q$ be the scalar projection of vector $\vec{BC}$ on vector $\vec{BA}$. If $\{x\}$ denotes the fractional part function, find the value of $18 \{ p \} + 9 \{ q \}$.

Let $\vec{u}(t) = (t^2 - 5t + 6)\hat{i} + (t^2 - 3t + 2)\hat{j}$ and $\vec{v}(t) = \hat{i} - \hat{j}$. Define a vector function $\vec{r}(t) = \text{sgn}(\vec{u}(t) \cdot \vec{v}(t)) \hat{i} + (|t-4| - 2) \hat{j}$ and let $\vec{w}(t) = (t-5) \hat{i} + \hat{j}$, where $\text{sgn}(\cdot)$ denotes the signum function. A function $F(x)$ is defined for $x > 0$ as: $F(x) = \int_{0}^{x} \vec{r}(t) \cdot \vec{w}(t) \, dt$ If $x_1$ and $x_2$ are the points of local minimum and local maximum of $F(x)$ respectively, find the value of $4(F(x_2) - F(x_1))$.

Let $u$ and $v$ be two unit vectors in $\mathbb{R}^3$ such that the angle between them is $\frac{\pi}{6}$. Let $P = uu^T$ and $Q = vv^T$ be the orthogonal projection matrices onto the one-dimensional subspaces spanned by $u$ and $v$, respectively. Define the transformation matrix $T = PQ - QP$. Which of the following represents the minimal polynomial $m(\lambda)$ of $T$?

Let $\vec{v}_k = \frac{k}{2} \hat{i} + |k-5| \hat{j}$ and $\vec{u}_k = \hat{i} + (-1)^k \hat{j}$ be two vectors for $k \in \mathbb{N}$. If $P_k$ is the scalar projection of $\vec{v}_k$ onto $\vec{u}_k$, evaluate the value of: $\left[ \sum_{k=1}^{21} \sqrt{2} P_k \right] - \sum_{k=1}^{21} \left[ \sqrt{2} P_k \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $\vec{u}$ and $\vec{v}$ be two variable vectors in $\mathbb{R}^3$ such that $|\vec{u}| = 2$ and $|\vec{v}| = 3$. Consider the scalar function $f(\vec{u}, \vec{v}) = |\vec{u} \times \vec{v}|^2 - 3(\vec{u} \cdot \vec{v})^2 + 12\vec{u} \cdot (2\vec{v} + \vec{u} \times \vec{v})$ If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(\vec{u}, \vec{v})$ respectively, find the value of $M - m$.

Let $\vec{a} = l\hat{i} + m\hat{j} + n\hat{k}$ be a unit vector, where $l, m, n$ are its direction cosines. Let $\vec{b} = \hat{i} + \hat{j} + \hat{k}$. Let $S$ be the set of all possible values of $\vec{a} \cdot \vec{b}$ that satisfy the inequality $\frac{|\vec{a} \times \vec{b}|^2 - 2}{\vec{a} \cdot \vec{b}} \ge 1$ If $M$ and $m$ are the supremum and infimum of the set $S$ respectively, then the value of $m^2 + \frac{1}{M}$ is:

Let $\vec{r}(t)$ be a twice differentiable vector-valued function such that $\vec{r} \times \vec{r}' \neq \vec{0}$. Evaluate the indefinite integral: $\int \frac{ \left( |\vec{r}|^4 |\vec{r}'|^2 - |\vec{r}|^2 (\vec{r} \cdot \vec{r}')^2 \right) (\vec{r} \times \vec{r}'') - 2(\vec{r} \cdot \vec{r}') |\vec{r} \times \vec{r}'|^2 (\vec{r} \times \vec{r}') }{ |\vec{r}|^6 |\vec{r}'|^2 - |\vec{r}|^4 (\vec{r} \cdot \vec{r}')^2 } \, dt$ (where $C$ is an arbitrary scalar constant and $\vec{C}$ is an arbitrary constant vector)

Let $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ be non-collinear vectors in $\mathbb{R}^3$ such that $\sum_{i=1}^3 a_i^2 = p^2$ and $\sum_{i=1}^3 b_i^2 = q^2$. For a given real parameter $r$ satisfying $r > pq$, what is the minimum value of the expression $E = \sum_{1 \le i < j \le 3} (a_i b_j - a_j b_i)^2 + \frac{r^4}{\sum_{1 \le i < j \le 3} (a_i b_j - a_j b_i)^2}$

Let $L_1$ and $L_2$ be lines in $\mathbb{R}^3$ defined by $y = \sqrt{3}x, z = 0$ and $y = -\sqrt{3}x, z = 0$ respectively. Let $S$ be the set of all points $P(x,y,z)$ such that there exists a unique plane $\Pi_1$ containing $P$ and $L_1$, a unique plane $\Pi_2$ containing $P$ and $L_2$, and $\Pi_1$ is perpendicular to $\Pi_2$. Which of the following points does NOT belong to the set $S$?

Let $n \in \mathbb{N}$. Consider the vectors $\vec{a} = 2n(n+1)\hat{i}$ and $\vec{b} = 2n(n+1)\hat{i} + (2n+1)\hat{j}$. Let $L$ be the line passing through the origin and parallel to the internal angle bisector of $\vec{a}$ and $\vec{b}$. If the line $L$ intersects the plane $x = 4n(n+1)$ at the point $P(x_0, y_0, z_0)$, then the value of $\lfloor y_0 \rfloor$ is (where $\lfloor \cdot \rfloor$ denotes the greatest integer function):

Let $\vec{e}_1, \vec{e}_2, \vec{e}_3$ be the standard orthonormal basis of $\mathbb{R}^3$. Let $A$ and $B$ be two $3 \times 3$ orthogonal matrices. For $i \in \{1, 2, 3\}$, let $\vec{w}_i = AB\vec{e}_i + BA\vec{e}_i$ (which lies along the internal angle bisector of the vectors $AB\vec{e}_i$ and $BA\vec{e}_i$). If $S = \text{span}(\vec{w}_1, \vec{w}_2, \vec{w}_3)$, which of the following sets represents all possible values for the dimension of $S$?

Let $\vec{u}, \vec{v}, \vec{w}$ be three mutually orthogonal unit vectors in $\mathbb{R}^3$. Let $A$ be the $3 \times 3$ matrix whose rows are $\vec{u}^T, \vec{v}^T, \vec{w}^T$. Let $t = u_1 + v_2 + w_3$ and $\delta = \vec{u} \cdot (\vec{v} \times \vec{w})$. If $[\delta t] = k$ for $k \in \{1, 2\}$, where $[\cdot]$ denotes the greatest integer function, which of the following represents the exact range of $\text{tr}(A^2)$?

Let $P_1$ be a plane with normal vector $\vec{n}_1 = l\hat{i} + m\hat{j} + n\hat{k}$, where $l, m, n$ are direction cosines such that $n = \frac{1}{2}$. Let $P_2$ be the plane $\sqrt{3}x + y + 2\sqrt{3}z = 0$. If $M$ is the maximum possible value of $\cos \alpha$, where $\alpha$ is the acute angle between $P_1$ and $P_2$, and $x \in (0, \frac{\pi}{2})$ satisfies the equation $4M^2 \sin^2 x - 4M \sin x + 1 = 0$, then the value of $\sec^2 x$ is:

Let $\hat{u} = l\hat{i} + m\hat{j} + n\hat{k}$ be a unit vector such that $n=0$. A point $P(al, bm)$ lies on the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) and $\vec{v}$ is the position vector of $P$.

The points $P$ are constrained by the modulus inequality: $|\vec{v}| \le \sqrt{\frac{a^2+b^2}{2}}$

Let $Q(al, am)$ be the point on the auxiliary circle of $E$ corresponding to $P$. The region $R$ is the union of the sectors of the auxiliary circle subtended at the origin by all valid points $Q$.

If the area of $R$ is $18\pi$ and the eccentricity of $E$ is $\frac{\sqrt{5}}{3}$, find the area of the ellipse $E$.

Let $\vec{a}(t), \vec{b}(t), \vec{c}(t)$ be three non-coplanar differentiable vector functions of $t \in \mathbb{R}$ such that their scalar triple product $V(t) = [\vec{a}(t) \vec{b}(t) \vec{c}(t)] \ge 1$ for all $t \ge 0$, and $V(0) = 1$. Let $\vec{a}', \vec{b}', \vec{c}'$ be the reciprocal system of vectors corresponding to $\vec{a}, \vec{b}, \vec{c}$, and let $W(t) = [\vec{a}'(t) \vec{b}'(t) \vec{c}'(t)]$. Define the functions: $f(t) = \frac{d\vec{a}}{dt} \cdot \vec{a}' + \frac{d\vec{b}}{dt} \cdot \vec{b}' + \frac{d\vec{c}}{dt} \cdot \vec{c}'$ $g(t) = \left[ \frac{d\vec{a}'}{dt} \vec{b}' \vec{c}' \right] + \left[ \vec{a}' \frac{d\vec{b}'}{dt} \vec{c}' \right] + \left[ \vec{a}' \vec{b}' \frac{d\vec{c}'}{dt} \right]$ If $f(t)$ and $g(t)$ satisfy the relation $f(t) + g(t) = \frac{2t^3}{t^2+1} W(t)$ for all $t \ge 0$, find the value of the integral $\int_0^2 \left( f(t) + t W(t) \right) dt$.

Let $\vec{a} = \hat{i}$, $\vec{b} = x\hat{i} + \hat{j}$, and $\vec{c} = (x^2 - 3x + 1)\hat{i} + \hat{k}$ be three vectors for $x \in \mathbb{R}$. Let $\vec{a}', \vec{b}', \vec{c}'$ be the reciprocal system of vectors of $\vec{a}, \vec{b}, \vec{c}$. Let $f(x) = |\vec{a}'|^2 + |\vec{b}'|^2 + |\vec{c}'|^2$. If $x_1$ and $x_2$ (where $x_1 < x_2$) are the $x$-coordinates of the points of inflection of the curve $y = f(x)$, find the value of the scalar triple product $[\vec{a}'(x_1), \vec{a}'(x_2), \vec{a}'(x_1 + x_2)]$.

Let $f(x) = \int_0^x e^{-t^2} (t^2 - 4)(t - 3)^2 dt$ and $\lambda$ be the number of points where $f(x)$ attains a local extremum. Consider two skew lines $L_1$ and $L_2$ given by: $L_1: \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-2}{-1}$ $L_2: \frac{x+2}{1} = \frac{y-1}{2} = \frac{z-\mu}{1}$ If the shortest distance between $L_1$ and $L_2$ is at most $\lambda \sqrt{3}$, then the sum of all possible integer values of $\mu$ is:

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{c} = 3\vec{a} - \vec{b}$ and $\vec{a} \times \vec{b} = 2\hat{i} - \hat{j} + 2\hat{k}$. Let $A$ be a $3 \times 3$ matrix whose columns are the vectors $\vec{a} \times \vec{b}$, $\vec{b} \times \vec{c}$, and $\vec{c} \times \vec{a}$ respectively. Let $\alpha$ be the largest real eigenvalue of $A$. If $B = A^3 - 6A^2 + 2I$, where $I$ is the $3 \times 3$ identity matrix, find the value of $\det(B - \alpha I)$.

Let $\vec{u} = x \hat{i}$ and $\vec{v} = y \hat{j}$ (where $x, y > 0$) be two vectors. The line passing through the endpoints of $\vec{u}$ and $\vec{v}$ is tangent to the auxiliary circle of the ellipse $\frac{X^2}{9} + \frac{Y^2}{4} = 1$. Let $S$ be the area of the parallelogram defined by $\vec{u}$ and $\vec{v}$. Find the sum of all possible integer values of $\frac{x^2}{y^2}$ that satisfy the inequality $\frac{S - 36}{S - 18} \le 0$

Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors in $\mathbb{R}^3$. A scalar field $f(\vec{r})$ with $f(\vec{0}) = 0$ has its gradient given by $\nabla f(\vec{r}) = g(\vec{r} \cdot (\vec{a} \times \vec{b})) (\vec{a} \times \vec{b})$ where $g(x) = \int_0^x (1 - 2\{t\}) dt$ and $\{t\}$ denotes the fractional part of $t$. Find the maximum value of $f(\vec{r})$ in the spherical region $|\vec{r}| \le \frac{n}{|\vec{a} \times \vec{b}|}$, where $n$ is a positive integer.

Let $z_1, z_2, z_3$ be the complex roots of the polynomial equation $z^3 - \alpha z^2 + \beta z - \gamma = 0$, where $\alpha, \beta, \gamma \in \mathbb{C}$. Consider the three vectors in $\mathbb{R}^3$: $\vec{v}_1 = |z_1|\hat{i} + |z_2|\hat{j} + |z_3|\hat{k}$ $\vec{v}_2 = |z_2|\hat{i} + |z_3|\hat{j} + |z_1|\hat{k}$ $\vec{v}_3 = |z_3|\hat{i} + |z_1|\hat{j} + |z_2|\hat{k}$ The volume of the parallelepiped formed by $\vec{v}_1, \vec{v}_2, \vec{v}_3$ is zero. If $|\alpha| = 6$, $\text{Re}(\alpha) = 3$, $\text{Im}(\alpha) > 0$, and the roots satisfy the condition $|\alpha| = |z_1| + |z_2| + |z_3|$, find the value of $|\beta - \gamma|^2$.

Consider the family of conics $C_\lambda : \frac{x^2}{\lambda} + \frac{y^2}{1-\lambda} = 1$, where $\lambda \in \mathbb{R} \setminus \{0, 1\}$. Let $D_\lambda$ be the director circle of $C_\lambda$. Let $\Gamma_\lambda$ be the locus of points $P$ such that the chord of contact of tangents from $P$ to $C_\lambda$ subtends a right angle at the origin. Let $S$ be the set of all $\lambda$ for which both $D_\lambda$ and $\Gamma_\lambda$ are real curves containing at least one point. For $\lambda \in S$, let $e_1$ and $e_2$ be the eccentricities of $C_\lambda$ and $\Gamma_\lambda$ respectively. Which of the following is true?

Consider three lines in $\mathbb{R}^3$ given by: $L_1: \vec{r} = 119\hat{i} + t(\hat{j} + \hat{k})$ $L_2: \vec{r} = 239\hat{j} + t\hat{k}$ $L_3: \vec{r} = \hat{k} + t\hat{i}$ Let $A, B,$ and $C$ be points on $L_1, L_2,$ and $L_3$ respectively, corresponding to the same parametric value $t \in \mathbb{R}$. Let $O$ be the origin. If the scalar triple product $S(t) = [\vec{OA}, \vec{OB}, \vec{OC}]$ attains a local minimum at $t = t_0 > 0$, find the value of $3t_0 - 239$.

Let $\alpha = \sin^{-1}\left(\sin \frac{10\pi}{9}\right)$ and $\beta = \cos^{-1}\left(\cos \frac{10\pi}{9}\right)$. Let $A_1, A_2, \dots, A_8$ be $8$ arithmetic means inserted between $\alpha$ and $\beta$. Let $d$ be the common difference of this arithmetic progression, and define $k = \frac{\pi}{d}$. Consider three vectors $\vec{u}, \vec{v}, \vec{w}$ defined as: $\vec{u} = \sin^{-1}(\sin A_{k-5})\hat{i} + \cos^{-1}(\cos A_{k-3})\hat{j} + \tan^{-1}(\tan A_{k-1})\hat{k}$ $\vec{v} = \cos^{-1}(\cos A_{k-1})\hat{i} + \tan^{-1}(\tan A_{k-5})\hat{j} + \sin^{-1}(\sin A_{k-3})\hat{k}$ $\vec{w} = \tan^{-1}(\tan A_{k-3})\hat{i} + \sin^{-1}(\sin A_{k-1})\hat{j} + \cos^{-1}(\cos A_{k-5})\hat{k}$ Find the volume of the parallelepiped whose coterminous edges are represented by $\vec{u}, \vec{v}$, and $\vec{w}$.