87903 If $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are three vectors, such that $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}.|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3,$ then $\overrightarrow{\mathbf{a}}.\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}}.\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}}.\overrightarrow{a}$ is equal to

87904 Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ 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 $164{\cos }^2\theta$ is equal to :

87905 Let âa and be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$, then among the statements:
$\left({\mathrm{S}}_1\right):2|\hat{\mathrm{a}}\times \hat{\mathrm{b}}|=|\hat{\mathrm{a}}-\hat{\mathrm{b}}|$
$\left(S_2\right)$ : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

87903 If $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are three vectors, such that $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}.|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3,$ then $\overrightarrow{\mathbf{a}}.\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}}.\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}}.\overrightarrow{a}$ is equal to

87904 Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ 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 $164{\cos }^2\theta$ is equal to :

87905 Let âa and be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$, then among the statements:
$\left({\mathrm{S}}_1\right):2|\hat{\mathrm{a}}\times \hat{\mathrm{b}}|=|\hat{\mathrm{a}}-\hat{\mathrm{b}}|$
$\left(S_2\right)$ : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

87906 Let $\hat{a},\hat{b}$ be unit vectors. If $\hat{\mathbf{c}}$ be a vector such that the angle between $\hat{a}$ and $\overrightarrow{c}$ is $\frac{\pi }{12}$, and $\hat{\mathbf{b}}=\overrightarrow{\mathbf{c}}+2(\overrightarrow{\mathbf{c}}\times \hat{\mathbf{a}})$ then $|\mathbf{6}{|}^2$ is equal to

87903 If $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are three vectors, such that $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}.|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3,$ then $\overrightarrow{\mathbf{a}}.\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}}.\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}}.\overrightarrow{a}$ is equal to

87904 Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ 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 $164{\cos }^2\theta$ is equal to :

87905 Let âa and be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$, then among the statements:
$\left({\mathrm{S}}_1\right):2|\hat{\mathrm{a}}\times \hat{\mathrm{b}}|=|\hat{\mathrm{a}}-\hat{\mathrm{b}}|$
$\left(S_2\right)$ : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

87906 Let $\hat{a},\hat{b}$ be unit vectors. If $\hat{\mathbf{c}}$ be a vector such that the angle between $\hat{a}$ and $\overrightarrow{c}$ is $\frac{\pi }{12}$, and $\hat{\mathbf{b}}=\overrightarrow{\mathbf{c}}+2(\overrightarrow{\mathbf{c}}\times \hat{\mathbf{a}})$ then $|\mathbf{6}{|}^2$ is equal to

87903 If $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are three vectors, such that $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}.|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3,$ then $\overrightarrow{\mathbf{a}}.\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}}.\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}}.\overrightarrow{a}$ is equal to

87904 Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ 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 $164{\cos }^2\theta$ is equal to :