Angle Between Two Vectors

Find the magnitude of each of the two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, having the same magnitude such that the angle between them is $60°$ and their scalar product is $\frac{9}{2}$.

Write the angle between two vectors $\overrightarrow{\mathit{a}}$and $\overrightarrow{\mathrm{b}}$ with magnitudes$\sqrt{3}$ and$2$ respectively having $\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}=\sqrt{6}$

Find the angle between the vectors $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$.

Find the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, if $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=3\text{ and }\overrightarrow{a}·\overrightarrow{b}=1$.

Find the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $\overrightarrow{a}=3\hat{i}-2\hat{j}-6\hat{k}$ and $\overrightarrow{b}=4\hat{i}-\hat{j}+8\hat{k}$

Find the angles of a triangle the coordinates of whose vertices are $A(0,-1,-2), B(3, 1, 4)$ and $(5, 7, 1)$.

If the vertices $A, B$ and $C$ of $△ABC$ have position vectors $(1, 2, 3), (-1, 0, 0)$and $(0, 1, 2)$, respectively, what is the value of $k$ if the magnitude of $\angle ABC={\cos }^{-1}\left(\frac{10}{\sqrt{k}}\right)$?

Find the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, if $\left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=2\text{ and }\overrightarrow{a}·\overrightarrow{b}=\sqrt{6}$.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, then find the value of $k$, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\mathrm{\pi }}{k}$, given that $\left(\sqrt{3}\overrightarrow{a}-\overrightarrow{b}\right)$ is a unit vector.

 If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+\overrightarrow{b}$ is also a unit vector, then find the value of $k$, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{k\mathrm{\pi }}{3}$

If $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are unit vectors, find the value of $k$ if the angle between $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ is $k°$

Find the value of $x+y$, if the cosine of the angle between the vectors  $4\hat{i}-3\hat{j}+3\hat{k}$ and $2\hat{i}-\hat{j}-\hat{k}$ is $\cos \theta =\frac{x}{\sqrt{y}}$.

What is the value of  $k$, if angle between vectors $\overrightarrow{a }and \overrightarrow{b}$ with magnitudes $2$ and $\sqrt{3}$ respectively is $\frac{\mathrm{\pi }}{k}$? Given $\overrightarrow{a}·\overrightarrow{b}=\sqrt{3}$

If $\overrightarrow{a}=\hat{i}-\hat{j}$ and $\overrightarrow{b}=-\hat{j}+2\hat{k}$, find $(\overrightarrow{a}-2\overrightarrow{b})·(\overrightarrow{a}+\overrightarrow{b})$.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors such that $\left|\overrightarrow{a}\right|=4, \left|\overrightarrow{b}\right|=3$ and $\overrightarrow{a}·\overrightarrow{b}=6$.  Find the value of $k$, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$is $\frac{\mathrm{\pi }}{k}$

Find the magnitude of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that are of the same magnitude, are inclined at $30°$ and whose scalar product is $\frac{1}{2}$.

Find the angles which the vector $\overrightarrow{a}=\hat{i}-\hat{j}+\sqrt{2}\hat{k}$ makes with the coordinate axes.

Find the value of $a+b$, if the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${\cos }^{-1}\left(\frac{-\sqrt{a}}{b}\right)$, where $\overrightarrow{a}=\hat{i}+2\hat{j}-\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$.

Find the value of $\frac{b}{a}$ if the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${\cos }^{-1}\left(\frac{-a}{\sqrt{b}}\right)$, where $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}-2\hat{k}$.

Find the value of $k$, if the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\mathrm{\pi }}{k}$, where $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=4\hat{i}+4\hat{j}-2\hat{k}$.