87903 If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are three vectors, such that \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}} .|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3, \quad\) then \(\overrightarrow{\mathbf{a}} . \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} . \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} . \vec{a}\) 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 \(\vec{c}\) 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 \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}} .|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3, \quad\) then \(\overrightarrow{\mathbf{a}} . \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} . \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} . \vec{a}\) 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 \(\vec{c}\) 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 \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}} .|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3, \quad\) then \(\overrightarrow{\mathbf{a}} . \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} . \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} . \vec{a}\) 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 \(\vec{c}\) 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 \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}} .|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=\mathbf{2},|\overrightarrow{\mathrm{c}}|=3, \quad\) then \(\overrightarrow{\mathbf{a}} . \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} . \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} . \vec{a}\) 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 \(\vec{c}\) 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