Dot product

Let $\vec a=\alpha\hat i+3\hat j-\hat k$, $\vec b=3\hat i-\beta\hat j+4\hat k$ and $\vec c=\hat i+2\hat j-2\hat k$ where $\alpha,\beta\in\mathbf R$, be three vectors. If the projection of $\vec a$ on $\vec c$ is $\dfrac{10}3$ and $\vec b\times\vec c=-6\hat i+10\hat j+7\hat k$, then the value of $\alpha+\beta$ is equal to :

Let $\vec a=\hat i-2\hat j+3\hat k$, $\vec b=\hat i+\hat j+\hat k$ and $\vec c$ be a vector such that $\vec a+\left(\vec b\times\vec c\right)=\vec0$ and $\vec b\cdot\vec c=5$. Then, the value of $3\left(\vec c\cdot\vec a\right)$ is equal to ______.

Let ABC be a triangle such that $\overrightarrow{BC}=\vec a$, $\overrightarrow{CA}=\vec b$, $\overrightarrow{AB}=\vec c$, $\left|\vec a\right|=6\sqrt2$, $\left|\vec b\right|=2\sqrt3$ and $\vec b\cdot\vec c=12$. Consider the statements :

(S1) : $\left|\left(\vec a\times\vec b\right)+\left(\vec c\times\vec b\right)\right|-\left|\vec c\right|=6\left(2\sqrt2-1\right)$

(S2) : $\angle ACB=\cos^{-1}\left(\sqrt{\dfrac23}\right)$

Then

Let $\vec a=\alpha\hat i+\hat j-\hat k$ and $\vec b=2\hat i+\hat j-\alpha\hat k$, $\alpha>0$. If the projection of $\vec a\times\vec b$ on the vector $-\hat i+2\hat j-2\hat k$ is 30, then $\alpha$ is equal to :

A vector $\vec a$ is parallel to the line of intersection of the plane determined by the vectors $\hat i$, $\hat i+\hat j$ and the plane determined by the vectors $\hat i-\hat j$, $\hat i+\hat k$. The obtuse angle between $\vec a$ and the vector $\vec b=\hat i-2\hat j+2\hat k$ is

Let $\vec a=\alpha\hat i+\hat j+\beta\hat k$ and $\vec b=3\hat i-5\hat j+4\hat k$ be two vectors, such that $\vec a\times\vec b=-\hat i+9\hat j+12\hat k$. Then the projection of $\vec b-2\vec a$ on $\vec b+\vec a$ is equal to

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 $P(-2,-1,1)$ and $Q\left(\dfrac{56}{17},\dfrac{43}{17},\dfrac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2+\beta^2$ 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 $\hat a$ 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 _______.

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 $\vec a = 2\hat i + \hat j - \hat k$ 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 ______