87956 If \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular vectors having magnitudes \(1,2,3\) respectively, then \(\left[\begin{array}{ll}\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}} \mathbf{c}\end{array}\right]=\) ?

87957 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=10\), then \(\lambda\) is equal to

87958 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\hat{\mathbf{k}}\), and \(\mathbf{c}=\mathbf{m} \overrightarrow{\mathbf{a}}+\mathbf{n} \overrightarrow{\mathbf{b}}\) then \(\mathbf{m}+\mathbf{n}=\)

87960 If \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\), then \(\sin \frac{\theta}{2}\) is

87956 If \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular vectors having magnitudes \(1,2,3\) respectively, then \(\left[\begin{array}{ll}\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}} \mathbf{c}\end{array}\right]=\) ?

87957 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=10\), then \(\lambda\) is equal to

87958 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\hat{\mathbf{k}}\), and \(\mathbf{c}=\mathbf{m} \overrightarrow{\mathbf{a}}+\mathbf{n} \overrightarrow{\mathbf{b}}\) then \(\mathbf{m}+\mathbf{n}=\)

87960 If \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\), then \(\sin \frac{\theta}{2}\) is

87956 If \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular vectors having magnitudes \(1,2,3\) respectively, then \(\left[\begin{array}{ll}\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}} \mathbf{c}\end{array}\right]=\) ?

87957 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=10\), then \(\lambda\) is equal to

87958 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\hat{\mathbf{k}}\), and \(\mathbf{c}=\mathbf{m} \overrightarrow{\mathbf{a}}+\mathbf{n} \overrightarrow{\mathbf{b}}\) then \(\mathbf{m}+\mathbf{n}=\)

87960 If \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\), then \(\sin \frac{\theta}{2}\) is

87956 If \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular vectors having magnitudes \(1,2,3\) respectively, then \(\left[\begin{array}{ll}\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}} \mathbf{c}\end{array}\right]=\) ?

87957 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=10\), then \(\lambda\) is equal to

87958 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\hat{\mathbf{k}}\), and \(\mathbf{c}=\mathbf{m} \overrightarrow{\mathbf{a}}+\mathbf{n} \overrightarrow{\mathbf{b}}\) then \(\mathbf{m}+\mathbf{n}=\)

87960 If \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\), then \(\sin \frac{\theta}{2}\) is