87749 If the points with position vectors \(-\hat{\mathbf{j}}-\hat{k}, 4 \hat{i}+5 \hat{j}+\lambda \hat{k}, 3 \hat{i}+9 \hat{j}+4 \hat{k}\) and \(-4 \hat{i}+4 \hat{j}+4 \hat{k}\) are coplanar, then \(\lambda\) equals

87750 If the vectors \(\overrightarrow{A B}=-3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(A B C\). Then the length of the median through \(A\) is

87751 If \(|\vec{a}|=2,|\vec{b}|=3,|\vec{c}|=4\) and \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\), then the value of \(\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}\) is equal to

87752 If the vectors \(\vec{i}+3 \vec{j}-2 \vec{k}, 2 \vec{i}-\vec{j}+4 \vec{k}\) and \(\mathbf{3} \overrightarrow{\mathbf{i}}+\mathbf{2} \vec{j}+\mathbf{x} \vec{k}\) are coplanar, then the value of \(x\) is:

87749 If the points with position vectors \(-\hat{\mathbf{j}}-\hat{k}, 4 \hat{i}+5 \hat{j}+\lambda \hat{k}, 3 \hat{i}+9 \hat{j}+4 \hat{k}\) and \(-4 \hat{i}+4 \hat{j}+4 \hat{k}\) are coplanar, then \(\lambda\) equals

87750 If the vectors \(\overrightarrow{A B}=-3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(A B C\). Then the length of the median through \(A\) is

87751 If \(|\vec{a}|=2,|\vec{b}|=3,|\vec{c}|=4\) and \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\), then the value of \(\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}\) is equal to

87752 If the vectors \(\vec{i}+3 \vec{j}-2 \vec{k}, 2 \vec{i}-\vec{j}+4 \vec{k}\) and \(\mathbf{3} \overrightarrow{\mathbf{i}}+\mathbf{2} \vec{j}+\mathbf{x} \vec{k}\) are coplanar, then the value of \(x\) is:

87749 If the points with position vectors \(-\hat{\mathbf{j}}-\hat{k}, 4 \hat{i}+5 \hat{j}+\lambda \hat{k}, 3 \hat{i}+9 \hat{j}+4 \hat{k}\) and \(-4 \hat{i}+4 \hat{j}+4 \hat{k}\) are coplanar, then \(\lambda\) equals

87750 If the vectors \(\overrightarrow{A B}=-3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(A B C\). Then the length of the median through \(A\) is

87751 If \(|\vec{a}|=2,|\vec{b}|=3,|\vec{c}|=4\) and \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\), then the value of \(\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}\) is equal to

87752 If the vectors \(\vec{i}+3 \vec{j}-2 \vec{k}, 2 \vec{i}-\vec{j}+4 \vec{k}\) and \(\mathbf{3} \overrightarrow{\mathbf{i}}+\mathbf{2} \vec{j}+\mathbf{x} \vec{k}\) are coplanar, then the value of \(x\) is:

87749 If the points with position vectors \(-\hat{\mathbf{j}}-\hat{k}, 4 \hat{i}+5 \hat{j}+\lambda \hat{k}, 3 \hat{i}+9 \hat{j}+4 \hat{k}\) and \(-4 \hat{i}+4 \hat{j}+4 \hat{k}\) are coplanar, then \(\lambda\) equals

87750 If the vectors \(\overrightarrow{A B}=-3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(A B C\). Then the length of the median through \(A\) is

87751 If \(|\vec{a}|=2,|\vec{b}|=3,|\vec{c}|=4\) and \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\), then the value of \(\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}\) is equal to

87752 If the vectors \(\vec{i}+3 \vec{j}-2 \vec{k}, 2 \vec{i}-\vec{j}+4 \vec{k}\) and \(\mathbf{3} \overrightarrow{\mathbf{i}}+\mathbf{2} \vec{j}+\mathbf{x} \vec{k}\) are coplanar, then the value of \(x\) is: