87784 Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-coplanar vectors and \(\vec{a}^{\prime}=\frac{\vec{b} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{b}^{\prime}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{c}}^{\prime} \frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot\) The length of the altitude of the parallelepiped formed by \(\vec{a}\) ', \(\overrightarrow{\mathbf{b}}^{\prime}, \overrightarrow{\mathbf{c}}^{\prime}\) as coterminous edges, with respect to the base having \(\overrightarrow{\mathbf{a}}{ }^{\prime}\) and \(\overrightarrow{\mathbf{c}}^{\prime}\) as its adjacent sides is

87787 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) are three vectors such that \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\) and \(\vec{b} \cdot \vec{c}=0\). If the projection of \(b\) along \(a\) is equal to the projection of \(\overrightarrow{\mathbf{c}}\) along \(\overrightarrow{\mathbf{a}}\), then \(|\mathbf{2} \overrightarrow{\mathbf{a}}+\mathbf{3} \overrightarrow{\mathbf{b}}-\mathbf{3} \overrightarrow{\mathbf{c}}|=\)

87788 For a non-zero real number \(x\), if the points with position vectors
\((x-u) \hat{i}+x \hat{j}+x \hat{k}, x \hat{i}+(x-v) \hat{j}+x \hat{k} \text {, }\)
\(x \hat{i}+x \hat{j}+(x-w) \hat{k} \text { and }\)
\((x-1) \hat{i}+(x-1) \hat{j}+(x-1) \hat{k}\) are coplanar, then

87789 Let \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) be two vectors in \(R^2\). If \(|\vec{u}+\overrightarrow{\mathbf{v}}|^2=\mathbf{2}\left(|\overrightarrow{\mathbf{u}}|^2+|\overrightarrow{\mathbf{v}}|^2\right)\), then ......

87790 If \(O\) is any point \(\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}+\overrightarrow{O D}=x \overrightarrow{O E}\), then find \(x\), given that \(A B C D\) is quadrilateral, \(E\) is the point of intersection of the line joining the mid-points of opposite sides.

87784 Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-coplanar vectors and \(\vec{a}^{\prime}=\frac{\vec{b} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{b}^{\prime}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{c}}^{\prime} \frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot\) The length of the altitude of the parallelepiped formed by \(\vec{a}\) ', \(\overrightarrow{\mathbf{b}}^{\prime}, \overrightarrow{\mathbf{c}}^{\prime}\) as coterminous edges, with respect to the base having \(\overrightarrow{\mathbf{a}}{ }^{\prime}\) and \(\overrightarrow{\mathbf{c}}^{\prime}\) as its adjacent sides is

87787 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) are three vectors such that \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\) and \(\vec{b} \cdot \vec{c}=0\). If the projection of \(b\) along \(a\) is equal to the projection of \(\overrightarrow{\mathbf{c}}\) along \(\overrightarrow{\mathbf{a}}\), then \(|\mathbf{2} \overrightarrow{\mathbf{a}}+\mathbf{3} \overrightarrow{\mathbf{b}}-\mathbf{3} \overrightarrow{\mathbf{c}}|=\)

87788 For a non-zero real number \(x\), if the points with position vectors
\((x-u) \hat{i}+x \hat{j}+x \hat{k}, x \hat{i}+(x-v) \hat{j}+x \hat{k} \text {, }\)
\(x \hat{i}+x \hat{j}+(x-w) \hat{k} \text { and }\)
\((x-1) \hat{i}+(x-1) \hat{j}+(x-1) \hat{k}\) are coplanar, then

87789 Let \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) be two vectors in \(R^2\). If \(|\vec{u}+\overrightarrow{\mathbf{v}}|^2=\mathbf{2}\left(|\overrightarrow{\mathbf{u}}|^2+|\overrightarrow{\mathbf{v}}|^2\right)\), then ......

87790 If \(O\) is any point \(\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}+\overrightarrow{O D}=x \overrightarrow{O E}\), then find \(x\), given that \(A B C D\) is quadrilateral, \(E\) is the point of intersection of the line joining the mid-points of opposite sides.

87784 Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-coplanar vectors and \(\vec{a}^{\prime}=\frac{\vec{b} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{b}^{\prime}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{c}}^{\prime} \frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot\) The length of the altitude of the parallelepiped formed by \(\vec{a}\) ', \(\overrightarrow{\mathbf{b}}^{\prime}, \overrightarrow{\mathbf{c}}^{\prime}\) as coterminous edges, with respect to the base having \(\overrightarrow{\mathbf{a}}{ }^{\prime}\) and \(\overrightarrow{\mathbf{c}}^{\prime}\) as its adjacent sides is

87787 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) are three vectors such that \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\) and \(\vec{b} \cdot \vec{c}=0\). If the projection of \(b\) along \(a\) is equal to the projection of \(\overrightarrow{\mathbf{c}}\) along \(\overrightarrow{\mathbf{a}}\), then \(|\mathbf{2} \overrightarrow{\mathbf{a}}+\mathbf{3} \overrightarrow{\mathbf{b}}-\mathbf{3} \overrightarrow{\mathbf{c}}|=\)

87788 For a non-zero real number \(x\), if the points with position vectors
\((x-u) \hat{i}+x \hat{j}+x \hat{k}, x \hat{i}+(x-v) \hat{j}+x \hat{k} \text {, }\)
\(x \hat{i}+x \hat{j}+(x-w) \hat{k} \text { and }\)
\((x-1) \hat{i}+(x-1) \hat{j}+(x-1) \hat{k}\) are coplanar, then

87789 Let \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) be two vectors in \(R^2\). If \(|\vec{u}+\overrightarrow{\mathbf{v}}|^2=\mathbf{2}\left(|\overrightarrow{\mathbf{u}}|^2+|\overrightarrow{\mathbf{v}}|^2\right)\), then ......

87790 If \(O\) is any point \(\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}+\overrightarrow{O D}=x \overrightarrow{O E}\), then find \(x\), given that \(A B C D\) is quadrilateral, \(E\) is the point of intersection of the line joining the mid-points of opposite sides.

87784 Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-coplanar vectors and \(\vec{a}^{\prime}=\frac{\vec{b} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{b}^{\prime}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{c}}^{\prime} \frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot\) The length of the altitude of the parallelepiped formed by \(\vec{a}\) ', \(\overrightarrow{\mathbf{b}}^{\prime}, \overrightarrow{\mathbf{c}}^{\prime}\) as coterminous edges, with respect to the base having \(\overrightarrow{\mathbf{a}}{ }^{\prime}\) and \(\overrightarrow{\mathbf{c}}^{\prime}\) as its adjacent sides is

87787 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) are three vectors such that \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\) and \(\vec{b} \cdot \vec{c}=0\). If the projection of \(b\) along \(a\) is equal to the projection of \(\overrightarrow{\mathbf{c}}\) along \(\overrightarrow{\mathbf{a}}\), then \(|\mathbf{2} \overrightarrow{\mathbf{a}}+\mathbf{3} \overrightarrow{\mathbf{b}}-\mathbf{3} \overrightarrow{\mathbf{c}}|=\)

87788 For a non-zero real number \(x\), if the points with position vectors
\((x-u) \hat{i}+x \hat{j}+x \hat{k}, x \hat{i}+(x-v) \hat{j}+x \hat{k} \text {, }\)
\(x \hat{i}+x \hat{j}+(x-w) \hat{k} \text { and }\)
\((x-1) \hat{i}+(x-1) \hat{j}+(x-1) \hat{k}\) are coplanar, then

87789 Let \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) be two vectors in \(R^2\). If \(|\vec{u}+\overrightarrow{\mathbf{v}}|^2=\mathbf{2}\left(|\overrightarrow{\mathbf{u}}|^2+|\overrightarrow{\mathbf{v}}|^2\right)\), then ......

87790 If \(O\) is any point \(\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}+\overrightarrow{O D}=x \overrightarrow{O E}\), then find \(x\), given that \(A B C D\) is quadrilateral, \(E\) is the point of intersection of the line joining the mid-points of opposite sides.

87784 Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-coplanar vectors and \(\vec{a}^{\prime}=\frac{\vec{b} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{b}^{\prime}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \overrightarrow{\mathbf{c}}^{\prime} \frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot\) The length of the altitude of the parallelepiped formed by \(\vec{a}\) ', \(\overrightarrow{\mathbf{b}}^{\prime}, \overrightarrow{\mathbf{c}}^{\prime}\) as coterminous edges, with respect to the base having \(\overrightarrow{\mathbf{a}}{ }^{\prime}\) and \(\overrightarrow{\mathbf{c}}^{\prime}\) as its adjacent sides is

87787 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) are three vectors such that \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\) and \(\vec{b} \cdot \vec{c}=0\). If the projection of \(b\) along \(a\) is equal to the projection of \(\overrightarrow{\mathbf{c}}\) along \(\overrightarrow{\mathbf{a}}\), then \(|\mathbf{2} \overrightarrow{\mathbf{a}}+\mathbf{3} \overrightarrow{\mathbf{b}}-\mathbf{3} \overrightarrow{\mathbf{c}}|=\)

87788 For a non-zero real number \(x\), if the points with position vectors
\((x-u) \hat{i}+x \hat{j}+x \hat{k}, x \hat{i}+(x-v) \hat{j}+x \hat{k} \text {, }\)
\(x \hat{i}+x \hat{j}+(x-w) \hat{k} \text { and }\)
\((x-1) \hat{i}+(x-1) \hat{j}+(x-1) \hat{k}\) are coplanar, then

87789 Let \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) be two vectors in \(R^2\). If \(|\vec{u}+\overrightarrow{\mathbf{v}}|^2=\mathbf{2}\left(|\overrightarrow{\mathbf{u}}|^2+|\overrightarrow{\mathbf{v}}|^2\right)\), then ......

87790 If \(O\) is any point \(\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}+\overrightarrow{O D}=x \overrightarrow{O E}\), then find \(x\), given that \(A B C D\) is quadrilateral, \(E\) is the point of intersection of the line joining the mid-points of opposite sides.