88198 \(l, \mathrm{~m}, \mathrm{n}\) are three unit vectors in a right handed system and \(L\) is a line through the points \(A, B\), C whose position vectors are \(p l+7 \mathrm{~m}-6 \mathrm{n}, 2 l+\) \(5 m-4 n\) and \(l+4 m-3 n\) respectively. If the equation of the plane containing \(L\) and the points \((-p, p, p+1)\) is \(\mathbf{a x}+\mathrm{by}+\mathrm{cz}=l\), then \(\mathrm{p}(\mathrm{a}\) \(+\mathbf{b}+\mathbf{c})=\)

88199 Let A, B, C be three points whose position vectors respectively are:
\(\vec{a}=\hat{i}+4 \hat{j}+3 \hat{k}\)
\(\overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}}, \alpha \in \square\)
\(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{5} \hat{\mathbf{k}}\)
If \(\alpha\) is the smallest positive integer for which \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear, then the length of the median, in \(\triangle \mathrm{ABC}\), through \(\mathrm{A}\) is :

88200 If \(a, b, c\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(a+2 b+3 c, \lambda b+\) \(4 c\) and \((2 \lambda-1) \mathrm{c}\) are non-coplanar for

88198 \(l, \mathrm{~m}, \mathrm{n}\) are three unit vectors in a right handed system and \(L\) is a line through the points \(A, B\), C whose position vectors are \(p l+7 \mathrm{~m}-6 \mathrm{n}, 2 l+\) \(5 m-4 n\) and \(l+4 m-3 n\) respectively. If the equation of the plane containing \(L\) and the points \((-p, p, p+1)\) is \(\mathbf{a x}+\mathrm{by}+\mathrm{cz}=l\), then \(\mathrm{p}(\mathrm{a}\) \(+\mathbf{b}+\mathbf{c})=\)

88199 Let A, B, C be three points whose position vectors respectively are:
\(\vec{a}=\hat{i}+4 \hat{j}+3 \hat{k}\)
\(\overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}}, \alpha \in \square\)
\(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{5} \hat{\mathbf{k}}\)
If \(\alpha\) is the smallest positive integer for which \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear, then the length of the median, in \(\triangle \mathrm{ABC}\), through \(\mathrm{A}\) is :

88200 If \(a, b, c\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(a+2 b+3 c, \lambda b+\) \(4 c\) and \((2 \lambda-1) \mathrm{c}\) are non-coplanar for

88198 \(l, \mathrm{~m}, \mathrm{n}\) are three unit vectors in a right handed system and \(L\) is a line through the points \(A, B\), C whose position vectors are \(p l+7 \mathrm{~m}-6 \mathrm{n}, 2 l+\) \(5 m-4 n\) and \(l+4 m-3 n\) respectively. If the equation of the plane containing \(L\) and the points \((-p, p, p+1)\) is \(\mathbf{a x}+\mathrm{by}+\mathrm{cz}=l\), then \(\mathrm{p}(\mathrm{a}\) \(+\mathbf{b}+\mathbf{c})=\)

88199 Let A, B, C be three points whose position vectors respectively are:
\(\vec{a}=\hat{i}+4 \hat{j}+3 \hat{k}\)
\(\overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}}, \alpha \in \square\)
\(\overrightarrow{\mathbf{c}}=\mathbf{3} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{5} \hat{\mathbf{k}}\)
If \(\alpha\) is the smallest positive integer for which \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear, then the length of the median, in \(\triangle \mathrm{ABC}\), through \(\mathrm{A}\) is :

88200 If \(a, b, c\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(a+2 b+3 c, \lambda b+\) \(4 c\) and \((2 \lambda-1) \mathrm{c}\) are non-coplanar for