Vector Algebra

A plane P is parallel to two lines whose direction ratios are −2, 1, −3 and −1, 2, −2 and it contains the point (2, 2, −2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts $\alpha$, $\beta$, $\gamma$. If V is the volume of the tetrahedron OABC, where O is the origin, and $p=\alpha+\beta+\gamma$, then the ordered pair (V, p) is equal to :

Let S be the set of all $a\in\mathbf R$ for which the angle between the vectors $\vec u=a(\log_eb)\hat i-6\hat j+3\hat k$ and $\vec v=(\log_eb)\hat i+2\hat j+2a(\log_eb)\hat k$, $(b>1)$ is acute. Then S is equal to :

Let $\vec a=3\hat i+\hat j$ and $\vec b=\hat i+2\hat j+\hat k$. Let $\vec c$ be a vector satisfying $\vec a\times(\vec b\times\vec c)=\vec b+\lambda\vec c$. If $\vec b$ and $\vec c$ are non-parallel, then the value of λ is :

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 :

The X-Y plane be taken as the boundary between two transparent media M$_1$ and M$_2$. M$_1$ in Z ⩾ 0 has a refractive index of $\sqrt2$ and M$_2$ with Z < 0 has a refractive index of $\sqrt3$. A ray of light travelling in M$_1$ along the direction given by the vector $\vec P=4\sqrt3\hat i-3\sqrt3\hat j-5\hat k$, is incident on the plane of separation. The value of difference between the angle of incident in M$_1$ and the angle of refraction in M$_2$ will be _______ degree.

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 OABC be a parallelogram, O be the origin, A (2, 4, −5) and C (b, 2, 3). If P (a, a, a) and Q (9 − a$^2$, 3, a − 1), a ∈ ℕ, are two points such that the projection of $\overrightarrow{OP}$ on $\overrightarrow{OB}$ is 2 and $\overrightarrow{OQ}$ makes acute angles with all three co-ordinate axes, then $|\overrightarrow{OB}|^2+|\overrightarrow{AC}|^2$ is equal to _______.

Let $\vec u$, $\vec v$ and $\vec w$ be vectors in three-dimensional space, where $\vec u$ and $\vec v$ are unit vectors which are not perpendicular to each other and
$$\vec u\cdot\vec w = 1,\qquad \vec v\cdot\vec w = 1,\qquad \vec w\cdot\vec w = 4$$
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $\vec u$, $\vec v$ and $\vec w$, is $\sqrt2$, then the value of $|3\vec u + 5\vec v|$ is ____.

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?

Let three vectors $\vec a$, $\vec b$ and $\vec c$ be such that $\vec c$ is coplanar with $\vec a$ and $\vec b$, $\vec a\cdot\vec c=7$ and $\vec b$ is perpendicular to $\vec c$, where $\vec a=-\hat i+\hat j+\hat k$ and $\vec b=2\hat i+\hat k$, then the value of $2|\vec a+\vec b+\vec c|^2$ is _______.

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 ______

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?

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 _____ .