87884 Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors such that \(\mathbf{c} \neq \mathbf{0}\) and \(\vec{a} \cdot \vec{b}=2 \vec{a} \cdot \vec{c},|\vec{a}|=|\vec{c}|=1,|\vec{b}|=4\) and \(|\vec{b} \times \vec{c}|=\) \(\sqrt{15}\), if \(b-2 c=\lambda \alpha\). then, \(\lambda\) equals

87885 If \(\vec{l}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \overrightarrow{\mathrm{m}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{n}}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}}\) are three vectors, then what is the value of \(\lambda\) such that \(\vec{n}\) is perpendicular to \(\vec{\lambda}+\overrightarrow{\mathbf{m}}\) ?

87886 The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then, which one of the following gives possible values of \(\alpha\) and \(\beta\) ?

87889 \(\vec{a}=\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \vec{b}=\frac{1}{7}(3 \hat{i}-\lambda \hat{j}+2 \hat{k})\). If \(a\) and
\(b\) are mutually perpendicular, then value of \(\lambda\) is

87884 Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors such that \(\mathbf{c} \neq \mathbf{0}\) and \(\vec{a} \cdot \vec{b}=2 \vec{a} \cdot \vec{c},|\vec{a}|=|\vec{c}|=1,|\vec{b}|=4\) and \(|\vec{b} \times \vec{c}|=\) \(\sqrt{15}\), if \(b-2 c=\lambda \alpha\). then, \(\lambda\) equals

87885 If \(\vec{l}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \overrightarrow{\mathrm{m}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{n}}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}}\) are three vectors, then what is the value of \(\lambda\) such that \(\vec{n}\) is perpendicular to \(\vec{\lambda}+\overrightarrow{\mathbf{m}}\) ?

87886 The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then, which one of the following gives possible values of \(\alpha\) and \(\beta\) ?

87889 \(\vec{a}=\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \vec{b}=\frac{1}{7}(3 \hat{i}-\lambda \hat{j}+2 \hat{k})\). If \(a\) and
\(b\) are mutually perpendicular, then value of \(\lambda\) is

87884 Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors such that \(\mathbf{c} \neq \mathbf{0}\) and \(\vec{a} \cdot \vec{b}=2 \vec{a} \cdot \vec{c},|\vec{a}|=|\vec{c}|=1,|\vec{b}|=4\) and \(|\vec{b} \times \vec{c}|=\) \(\sqrt{15}\), if \(b-2 c=\lambda \alpha\). then, \(\lambda\) equals

87885 If \(\vec{l}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \overrightarrow{\mathrm{m}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{n}}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}}\) are three vectors, then what is the value of \(\lambda\) such that \(\vec{n}\) is perpendicular to \(\vec{\lambda}+\overrightarrow{\mathbf{m}}\) ?

87886 The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then, which one of the following gives possible values of \(\alpha\) and \(\beta\) ?

87889 \(\vec{a}=\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \vec{b}=\frac{1}{7}(3 \hat{i}-\lambda \hat{j}+2 \hat{k})\). If \(a\) and
\(b\) are mutually perpendicular, then value of \(\lambda\) is

87884 Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors such that \(\mathbf{c} \neq \mathbf{0}\) and \(\vec{a} \cdot \vec{b}=2 \vec{a} \cdot \vec{c},|\vec{a}|=|\vec{c}|=1,|\vec{b}|=4\) and \(|\vec{b} \times \vec{c}|=\) \(\sqrt{15}\), if \(b-2 c=\lambda \alpha\). then, \(\lambda\) equals

87885 If \(\vec{l}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \overrightarrow{\mathrm{m}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{n}}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}}\) are three vectors, then what is the value of \(\lambda\) such that \(\vec{n}\) is perpendicular to \(\vec{\lambda}+\overrightarrow{\mathbf{m}}\) ?

87886 The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then, which one of the following gives possible values of \(\alpha\) and \(\beta\) ?

87889 \(\vec{a}=\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \vec{b}=\frac{1}{7}(3 \hat{i}-\lambda \hat{j}+2 \hat{k})\). If \(a\) and
\(b\) are mutually perpendicular, then value of \(\lambda\) is