Dot Product of Two 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

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are position vectors of the points $\left(\alpha ,3,0\right)$ and $\left(1,0,0\right)$ respectively and if the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{4}$, then the value of $\alpha$ is equal to

If $\left|\overrightarrow{a}\right|=10$ and $\left|\overrightarrow{b}\right|=5$, then the value of $\left(\overrightarrow{a}+2\overrightarrow{b}\right)·\left(\overrightarrow{a}-2\overrightarrow{b}\right)$ is equal to

If $\left|\overrightarrow{a}\right|=\sqrt{14},\left|\overrightarrow{b}\right|=\sqrt{10},\left|\overrightarrow{a}-\overrightarrow{b}\right|=\sqrt{24}$ and $\theta$ is angle between $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\cos \theta =$

Let $\overrightarrow{a}=\hat{ı}+2\hat{ȷ}-3\hat{k}$ and $\overrightarrow{a}+\overrightarrow{b}=4\hat{ı}-2\hat{ȷ}+\lambda \hat{k}$. If $\overrightarrow{a}·\overrightarrow{b}=4$, then the value of $\lambda$ is equal to

If $\overrightarrow{e_1} \& \overrightarrow{e_2}$ are two unit vectors and $\theta$ is the angle between them, then $\sin \left(\frac{\theta }{2}\right)$ is -

Let $\hat{i}, \hat{j}$ and $\hat{k}$ denote the standard unit vectors in $R^3$ along the $x$-axis, $y$-axis and $z$-axis, respectively. Consider the sets

$X=\left\{a\hat{i}+b\hat{j}+c\hat{k} : a,b,c\in \left\{-1,0,1\right\}\right\}$ and

$Y=${$\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right) :\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\in X$ and $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$ are mutually perpendicular unit vectors}

Then, the number of elements in $Y$ is

$\overrightarrow{OA}=4\hat{i}+3\hat{j}+5\hat{k}$, $\overrightarrow{OB}=2\hat{i}+3\hat{j}+2\hat{k}$ and $\overrightarrow{BA}=\hat{i}+3\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=4\hat{i}+3\hat{j}+5\hat{k}$, $\overrightarrow{OB}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{BA}=3\hat{i}+\hat{j}+2\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=3\hat{i}+2\hat{j}+5\hat{k}$, $\overrightarrow{OB}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{BA}=2\hat{i}+2\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=3\hat{i}+2\hat{j}+5\hat{k}$, $\overrightarrow{OB}=4\hat{i}+5\hat{j}+2\hat{k}$ and $\overrightarrow{BA}=-\hat{i}-3\hat{j}+3\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.