87890 The vector equation of the plane through the point \(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and perpendicular to the line of intersection of the plane \(r .(3 \hat{i}-\hat{j}+\hat{k})=1\) and \(\mathbf{r} \cdot(\hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})=\mathbf{2}\) is

87891 A vector perpendicular to the plane containing the points \(A(1,-1,2), B(2,0,-1), C(0,2,1)\) is

87892 a, b, c are three vectors such that \(|a|=3,|b|=5\), \(|c|=7\). If \(a, b, c\) are perpendicular to the vectors \(\mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}, \mathbf{a}+\mathbf{b}\) respectively, then \(\sqrt{(a+b+c)^2}-2=\)

87893 The set of real values of \(\lambda\) for which the vectors \(\lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) and \(2 \lambda \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}-\hat{\mathbf{k}}\) are perpendicular to each other is

87890 The vector equation of the plane through the point \(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and perpendicular to the line of intersection of the plane \(r .(3 \hat{i}-\hat{j}+\hat{k})=1\) and \(\mathbf{r} \cdot(\hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})=\mathbf{2}\) is

87891 A vector perpendicular to the plane containing the points \(A(1,-1,2), B(2,0,-1), C(0,2,1)\) is

87892 a, b, c are three vectors such that \(|a|=3,|b|=5\), \(|c|=7\). If \(a, b, c\) are perpendicular to the vectors \(\mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}, \mathbf{a}+\mathbf{b}\) respectively, then \(\sqrt{(a+b+c)^2}-2=\)

87893 The set of real values of \(\lambda\) for which the vectors \(\lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) and \(2 \lambda \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}-\hat{\mathbf{k}}\) are perpendicular to each other is

87890 The vector equation of the plane through the point \(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and perpendicular to the line of intersection of the plane \(r .(3 \hat{i}-\hat{j}+\hat{k})=1\) and \(\mathbf{r} \cdot(\hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})=\mathbf{2}\) is

87891 A vector perpendicular to the plane containing the points \(A(1,-1,2), B(2,0,-1), C(0,2,1)\) is

87892 a, b, c are three vectors such that \(|a|=3,|b|=5\), \(|c|=7\). If \(a, b, c\) are perpendicular to the vectors \(\mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}, \mathbf{a}+\mathbf{b}\) respectively, then \(\sqrt{(a+b+c)^2}-2=\)

87893 The set of real values of \(\lambda\) for which the vectors \(\lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) and \(2 \lambda \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}-\hat{\mathbf{k}}\) are perpendicular to each other is

87890 The vector equation of the plane through the point \(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and perpendicular to the line of intersection of the plane \(r .(3 \hat{i}-\hat{j}+\hat{k})=1\) and \(\mathbf{r} \cdot(\hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})=\mathbf{2}\) is

87891 A vector perpendicular to the plane containing the points \(A(1,-1,2), B(2,0,-1), C(0,2,1)\) is

87892 a, b, c are three vectors such that \(|a|=3,|b|=5\), \(|c|=7\). If \(a, b, c\) are perpendicular to the vectors \(\mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}, \mathbf{a}+\mathbf{b}\) respectively, then \(\sqrt{(a+b+c)^2}-2=\)

87893 The set of real values of \(\lambda\) for which the vectors \(\lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) and \(2 \lambda \hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}-\hat{\mathbf{k}}\) are perpendicular to each other is