87682 If the area of the parallelogram with \(\vec{a}\) and \(\vec{b}\) as two adjacent sides is 15 sq. units then the area of the parallelogram having \(\mathbf{3 a}+\mathbf{2} \vec{b}\) and \(\vec{a}+3 \vec{b}\) as two adjacent sides in sq. units is

87683 The diagonals of a parallelogram are the vectors \(3 \hat{i}+6 \hat{j}-2 \hat{k}\) and \(-\hat{i}-2 \hat{j}-8 \hat{k}\), then the length of the shorter side of parallelogram is

87684 Let \((3,4,-1)\) and \((-1,2,3)\) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to

87685 Let \(\overrightarrow{\mathrm{r}}\) be the position vector of a point \(P(x, y, z)\), where \(x, y\) and \(z\) are natural numbers and \(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\). What is the total number of possible positions of point \(P\) for which \(\vec{r} . \vec{a}=10\) ?

87682 If the area of the parallelogram with \(\vec{a}\) and \(\vec{b}\) as two adjacent sides is 15 sq. units then the area of the parallelogram having \(\mathbf{3 a}+\mathbf{2} \vec{b}\) and \(\vec{a}+3 \vec{b}\) as two adjacent sides in sq. units is

87683 The diagonals of a parallelogram are the vectors \(3 \hat{i}+6 \hat{j}-2 \hat{k}\) and \(-\hat{i}-2 \hat{j}-8 \hat{k}\), then the length of the shorter side of parallelogram is

87684 Let \((3,4,-1)\) and \((-1,2,3)\) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to

87685 Let \(\overrightarrow{\mathrm{r}}\) be the position vector of a point \(P(x, y, z)\), where \(x, y\) and \(z\) are natural numbers and \(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\). What is the total number of possible positions of point \(P\) for which \(\vec{r} . \vec{a}=10\) ?

87682 If the area of the parallelogram with \(\vec{a}\) and \(\vec{b}\) as two adjacent sides is 15 sq. units then the area of the parallelogram having \(\mathbf{3 a}+\mathbf{2} \vec{b}\) and \(\vec{a}+3 \vec{b}\) as two adjacent sides in sq. units is

87683 The diagonals of a parallelogram are the vectors \(3 \hat{i}+6 \hat{j}-2 \hat{k}\) and \(-\hat{i}-2 \hat{j}-8 \hat{k}\), then the length of the shorter side of parallelogram is

87684 Let \((3,4,-1)\) and \((-1,2,3)\) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to

87685 Let \(\overrightarrow{\mathrm{r}}\) be the position vector of a point \(P(x, y, z)\), where \(x, y\) and \(z\) are natural numbers and \(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\). What is the total number of possible positions of point \(P\) for which \(\vec{r} . \vec{a}=10\) ?

87682 If the area of the parallelogram with \(\vec{a}\) and \(\vec{b}\) as two adjacent sides is 15 sq. units then the area of the parallelogram having \(\mathbf{3 a}+\mathbf{2} \vec{b}\) and \(\vec{a}+3 \vec{b}\) as two adjacent sides in sq. units is

87683 The diagonals of a parallelogram are the vectors \(3 \hat{i}+6 \hat{j}-2 \hat{k}\) and \(-\hat{i}-2 \hat{j}-8 \hat{k}\), then the length of the shorter side of parallelogram is

87684 Let \((3,4,-1)\) and \((-1,2,3)\) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to

87685 Let \(\overrightarrow{\mathrm{r}}\) be the position vector of a point \(P(x, y, z)\), where \(x, y\) and \(z\) are natural numbers and \(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\). What is the total number of possible positions of point \(P\) for which \(\vec{r} . \vec{a}=10\) ?