87690 The position vectors of the points \(A, B, C\) and \(D\) are \(2 \hat{i}+4 \hat{k}, 5 \hat{i}+3 \sqrt{3} \hat{j}+4 \hat{k},-2 \sqrt{3} \hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{k}\) respectively. Then \(\overline{\mathbf{C D}}=\)

87691 The distance of the point having position vector \(-\hat{i}+2 \hat{j}+6 \hat{k}\) from the straight line passing through the point \((2,3,-4)\) and parallel to the vector, \(6 \hat{i}+3 \hat{j}-4 \hat{k}\) is

87692 Let \(A(3,0,-1), B(2,10,6)\) and \(C(1,2,1)\) be the vertices of a triangle and \(M\) be the mid-point of \(A C\). If \(G\) divides \(B M\) in the ratio \(2: 1\) then cos \((\angle \mathrm{GOA})(0\), being the origin) is equal to

87693 If \(C\) is the mid-point of \(A B\) and \(P\) is any point outside \(A B\), then

87694 A vector \(a\) has components \(3 p\) and 1 with respect to rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components \(p\) +1 and \(\sqrt{10}\), then a value of \(p\) is equal to

87690 The position vectors of the points \(A, B, C\) and \(D\) are \(2 \hat{i}+4 \hat{k}, 5 \hat{i}+3 \sqrt{3} \hat{j}+4 \hat{k},-2 \sqrt{3} \hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{k}\) respectively. Then \(\overline{\mathbf{C D}}=\)

87691 The distance of the point having position vector \(-\hat{i}+2 \hat{j}+6 \hat{k}\) from the straight line passing through the point \((2,3,-4)\) and parallel to the vector, \(6 \hat{i}+3 \hat{j}-4 \hat{k}\) is

87692 Let \(A(3,0,-1), B(2,10,6)\) and \(C(1,2,1)\) be the vertices of a triangle and \(M\) be the mid-point of \(A C\). If \(G\) divides \(B M\) in the ratio \(2: 1\) then cos \((\angle \mathrm{GOA})(0\), being the origin) is equal to

87693 If \(C\) is the mid-point of \(A B\) and \(P\) is any point outside \(A B\), then

87694 A vector \(a\) has components \(3 p\) and 1 with respect to rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components \(p\) +1 and \(\sqrt{10}\), then a value of \(p\) is equal to

87690 The position vectors of the points \(A, B, C\) and \(D\) are \(2 \hat{i}+4 \hat{k}, 5 \hat{i}+3 \sqrt{3} \hat{j}+4 \hat{k},-2 \sqrt{3} \hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{k}\) respectively. Then \(\overline{\mathbf{C D}}=\)

87691 The distance of the point having position vector \(-\hat{i}+2 \hat{j}+6 \hat{k}\) from the straight line passing through the point \((2,3,-4)\) and parallel to the vector, \(6 \hat{i}+3 \hat{j}-4 \hat{k}\) is

87692 Let \(A(3,0,-1), B(2,10,6)\) and \(C(1,2,1)\) be the vertices of a triangle and \(M\) be the mid-point of \(A C\). If \(G\) divides \(B M\) in the ratio \(2: 1\) then cos \((\angle \mathrm{GOA})(0\), being the origin) is equal to

87693 If \(C\) is the mid-point of \(A B\) and \(P\) is any point outside \(A B\), then

87694 A vector \(a\) has components \(3 p\) and 1 with respect to rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components \(p\) +1 and \(\sqrt{10}\), then a value of \(p\) is equal to

87690 The position vectors of the points \(A, B, C\) and \(D\) are \(2 \hat{i}+4 \hat{k}, 5 \hat{i}+3 \sqrt{3} \hat{j}+4 \hat{k},-2 \sqrt{3} \hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{k}\) respectively. Then \(\overline{\mathbf{C D}}=\)

87691 The distance of the point having position vector \(-\hat{i}+2 \hat{j}+6 \hat{k}\) from the straight line passing through the point \((2,3,-4)\) and parallel to the vector, \(6 \hat{i}+3 \hat{j}-4 \hat{k}\) is

87692 Let \(A(3,0,-1), B(2,10,6)\) and \(C(1,2,1)\) be the vertices of a triangle and \(M\) be the mid-point of \(A C\). If \(G\) divides \(B M\) in the ratio \(2: 1\) then cos \((\angle \mathrm{GOA})(0\), being the origin) is equal to

87693 If \(C\) is the mid-point of \(A B\) and \(P\) is any point outside \(A B\), then

87694 A vector \(a\) has components \(3 p\) and 1 with respect to rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components \(p\) +1 and \(\sqrt{10}\), then a value of \(p\) is equal to

87690 The position vectors of the points \(A, B, C\) and \(D\) are \(2 \hat{i}+4 \hat{k}, 5 \hat{i}+3 \sqrt{3} \hat{j}+4 \hat{k},-2 \sqrt{3} \hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{k}\) respectively. Then \(\overline{\mathbf{C D}}=\)

87691 The distance of the point having position vector \(-\hat{i}+2 \hat{j}+6 \hat{k}\) from the straight line passing through the point \((2,3,-4)\) and parallel to the vector, \(6 \hat{i}+3 \hat{j}-4 \hat{k}\) is

87692 Let \(A(3,0,-1), B(2,10,6)\) and \(C(1,2,1)\) be the vertices of a triangle and \(M\) be the mid-point of \(A C\). If \(G\) divides \(B M\) in the ratio \(2: 1\) then cos \((\angle \mathrm{GOA})(0\), being the origin) is equal to

87693 If \(C\) is the mid-point of \(A B\) and \(P\) is any point outside \(A B\), then

87694 A vector \(a\) has components \(3 p\) and 1 with respect to rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components \(p\) +1 and \(\sqrt{10}\), then a value of \(p\) is equal to