Dot Product of Vectors

Let  $\overrightarrow{a}=\widehat{i}+4\widehat{j}+2\widehat{k}, \overrightarrow{b}=3\widehat{i}-2\widehat{j}+7\widehat{k}$ and  $\overrightarrow{c}=2\widehat{i}-\widehat{j}+4\widehat{k}$ . Which of the following is representing a vector $\overrightarrow{p}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and also whose scalar product with vector $\overrightarrow{c}$ would be $\overrightarrow{p}.\overrightarrow{c}=18.$

Two projectiles are fired from the same point with the same speed at angles of projection $60°\mathrm{and}30°$ respectively. The correct statement is

Let a vector $\overrightarrow{a}=4\hat{i}-8\hat{j}+\hat{k}$ make angles $\alpha , \beta , \gamma$ with the positive directions of $x, y, z$ axes respectively.

What is $\cos 2\beta +\cos 2\gamma$ equal to?

Let a vector $\overrightarrow{a}=4\hat{i}-8\hat{j}+\hat{k}$ make angles $\alpha , \beta , \gamma$ with the positive directions of $x, y, z$ axes respectively.

What is $\cos \alpha$ equal to?

If the magnitudes of $\overrightarrow{a}, \overrightarrow{b} \text{and} \overrightarrow{a}+\overrightarrow{b}$ are respectively $3,4$ and $5$, then the magnitude of $\left(\overrightarrow{a}-\overrightarrow{b}\right)$ is

The vector $\overrightarrow{A}=\hat{i}+\hat{j}$ and $\overrightarrow{B}=\hat{i}-\hat{j}$. The angle between $\overrightarrow{A} \& \overrightarrow{B }=$

An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\overrightarrow{OP}=\overrightarrow{u}, \overrightarrow{OR}=\overrightarrow{v}$ and $\overrightarrow{OQ}=\alpha \overrightarrow{u}+\beta \overrightarrow{v}$, then $\alpha , {\beta }^2$, are the roots of the equation

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be three non-coplanar vectors and $\stackrel{\rightharpoonup }{d}$ be a non-zero vector which is perpendicular to $\left(\overrightarrow{a}+ \overrightarrow{b}+\overrightarrow{c}\right)$ and is represented as $\stackrel{\rightharpoonup }{d}=x\left(\overrightarrow{a}\times \overrightarrow{b}\right)+y\left(\overrightarrow{b}\times \overrightarrow{c}\right)+z\left(\overrightarrow{c}\times \overrightarrow{a}\right)$. Then 

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors and $\theta$ is the angle between them. If $\left|\overrightarrow{A}\times \overrightarrow{B}\right|=\sqrt{3}\left(\overrightarrow{A}·\overrightarrow{B}\right)$, then the value of $\theta$ is

If $\overrightarrow{a}$ is any non-zero vector, then the value of $\left(\overrightarrow{a}.\hat{i}\right)\hat{i}+\left(\overrightarrow{a}.\hat{j}\right)\hat{j}+\left(\overrightarrow{a}.\hat{k}\right)\hat{k}$ is

If $\overrightarrow{v_1}+\overrightarrow{v_2}$ is $\perp$ to $\overrightarrow{v_1}-\overrightarrow{v_2},$ then

$\left(2\hat{i}-3\hat{j}\right).\left(\hat{i}+\hat{k}\right)=$

If $\hat{a}$ and $\hat{b}$ are orthogonal vectors, then for any-non zero vector $\overrightarrow{r}$, the vector $\left(\overrightarrow{r}\times \hat{a}\right)$ equals (where, $\left[\overrightarrow{r} \overrightarrow{a} \overrightarrow{b}\right]\ne 0$

The component of vector $\overrightarrow{A}=2\hat{i}+3\hat{j}+\hat{k}$ in the direction of $3\hat{i}+4\hat{j}$ is