Dot product

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 _____ .

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point $S$ is such that

$$\overrightarrow{OP}\cdot\overrightarrow{OQ} + \overrightarrow{OR}\cdot\overrightarrow{OS} = \overrightarrow{OR}\cdot\overrightarrow{OP} + \overrightarrow{OQ}\cdot\overrightarrow{OS} = \overrightarrow{OQ}\cdot\overrightarrow{OR} + \overrightarrow{OP}\cdot\overrightarrow{OS}$$

Then the triangle $PQR$ has $S$ as its

Let $\vec a=2\hat i+\hat j-2\hat k$ and $\vec b=\hat i+\hat j$. Let $\vec c$ be a vector such that $|\vec c-\vec a|=3$, $\left|(\vec a\times\vec b)\times\vec c\right|=3$ and the angle between $\vec c$ and $\vec a\times\vec b$ be 30°. Then $\vec a\cdot\vec c$ is equal to :

If the vector $\vec b=3\hat j+4\hat k$ is written as the sum of a vector $\vec b_1$, parallel to $\vec a=\hat i+\hat j$ and a vector $\vec b_2$, perpendicular to $\vec a$, then $\vec b_1\times\vec b_2$ is equal to :

Let $\hat u=u_1\hat i+u_2\hat j+u_3\hat k$ be a unit vector in $\mathbb R^3$ and $\hat w=\dfrac1{\sqrt6}(\hat i+\hat j+2\hat k)$. Given that there exists a vector $\vec v$ in $\mathbb R^3$ such that $|\hat u\times\vec v|=1$ and $\hat w\cdot(\hat u\times\vec v)=1$. Which of the following statement(s) is(are) correct?

In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively $3\hat i+\hat j-\hat k$, $-\hat i+3\hat j+p\hat k$ and $5\hat i+q\hat j-4\hat k$, then the point (p, q) lies on a line :

Let $\Delta PQR$ be a triangle. Let $\vec a=\overrightarrow{QR}$, $\vec b=\overrightarrow{RP}$ and $\vec c=\overrightarrow{PQ}$. If $|\vec a|=12$, $|\vec b|=4\sqrt3$ and $\vec b\cdot\vec c=24$, then which of the following is (are) true?

Let $\vec a$ and $\vec b$ be two unit vectors such that $|\vec a+\vec b|=\sqrt3$. If $\vec c=\vec a+2\vec b+3(\vec a\times\vec b)$, then $2|\vec c|$ is equal to :

In a parallelogram ABCD, $|\vec{AB}|=a$, $|\vec{AD}|=b$ and $|\vec{AC}|=c$, then $\vec{DB}\cdot\vec{AB}$ has the value :

Let $\vec x$, $\vec y$ and $\vec z$ be three vectors each of magnitude $\sqrt2$ and the angle between each pair of them is $\frac\pi3$. If $\vec a$ is a nonzero vector perpendicular to $\vec x$ and $\vec y\times\vec z$ and $\vec b$ is a nonzero vector perpendicular to $\vec y$ and $\vec z\times\vec x$, then

Let $\vec a$, $\vec b$ and $\vec c$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac\pi3$. If $\vec a\times\vec b+\vec b\times\vec c=p\vec a+q\vec b+r\vec c$, where $p$, $q$ and $r$ are scalars, then the value of
$$\frac{p^2+2q^2+r^2}{q^2}$$
is

If $|\vec a|=2$, $|\vec b|=3$ and $|2\vec a-\vec b|=5$, then $|2\vec a+\vec b|$ equals :

If $|\vec c|^2=60$ and $\vec c\times\left(\hat i+2\hat j+5\hat k\right)=\vec0$, then a value of $\vec c\cdot\left(-7\hat i+2\hat j+3\hat k\right)$ is :

If $\hat x$, $\hat y$ and $\hat z$ are three unit vectors in three-dimensional space, then the minimum value of
$$|\hat x+\hat y|^2+|\hat y+\hat z|^2+|\hat z+\hat x|^2$$
is :

If $\vec x=3\hat i-6\hat j-\hat k$, $\vec y=\hat i+4\hat j-3\hat k$ and $\vec z=3\hat i-4\hat j-12\hat k$, then the magnitude of the projection of $\vec x\times\vec y$ on $\vec z$ is :

If $\vec a$, $\vec b$ and $\vec c$ are unit vectors satisfying $|\vec a-\vec b|^2+|\vec b-\vec c|^2+|\vec c-\vec a|^2=9$, then $|2\vec a+5\vec b+5\vec c|$ is

If $\vec a$ and $\vec b$ are vectors such that $|\vec a+\vec b|=\sqrt{29}$ and $\vec a\times(2\hat i+3\hat j+4\hat k)=(2\hat i+3\hat j+4\hat k)\times\vec b$, then a possible value of $(\vec a+\vec b)\cdot(-7\hat i+2\hat j+3\hat k)$ 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 $\vec a=-\hat i-\hat k$, $\vec b=-\hat i+\hat j$ and $\vec c=\hat i+2\hat j+3\hat k$ be three given vectors. If $\vec r$ is a vector such that $\vec r\times\vec b=\vec c\times\vec b$ and $\vec r\cdot\vec a=0$, then the value of $\vec r\cdot\vec b$ is