87708 Let \(L\) be a line passing through a point \(A\) and parallel to the vector \(2 \hat{i}+\hat{j}-2 \hat{k}\). Let \(-7 \hat{i}-5 \hat{j}+11 \hat{k}\) be the position vector of a point \(P\) on \(L\) such that \(|A P|=12\). Then, the position vector of \(A\) can be

87709 Let the vectors \(A B=2 \hat{i}+2 \hat{j}+\hat{k}\) and \(A C=2 \hat{i}+4 \hat{j}+4 \hat{k}\) be two sides of a \(\triangle A B C\). If \(G\) is the centroid of \(\triangle A B C\), then \(\frac{27}{7}|\mathrm{AG}|^2+5=\)

87710 \(\mathrm{L}_1\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}-\mathbf{3} \hat{\mathbf{k}}\). \(\mathrm{L}_2\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\). Then the distance between \(L_1\) and \(L_2\) is

87711 A vector a has components \(2 p\) and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If \(a\) has components \(p+1\) and 1 with respect to the new system, then

87708 Let \(L\) be a line passing through a point \(A\) and parallel to the vector \(2 \hat{i}+\hat{j}-2 \hat{k}\). Let \(-7 \hat{i}-5 \hat{j}+11 \hat{k}\) be the position vector of a point \(P\) on \(L\) such that \(|A P|=12\). Then, the position vector of \(A\) can be

87709 Let the vectors \(A B=2 \hat{i}+2 \hat{j}+\hat{k}\) and \(A C=2 \hat{i}+4 \hat{j}+4 \hat{k}\) be two sides of a \(\triangle A B C\). If \(G\) is the centroid of \(\triangle A B C\), then \(\frac{27}{7}|\mathrm{AG}|^2+5=\)

87710 \(\mathrm{L}_1\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}-\mathbf{3} \hat{\mathbf{k}}\). \(\mathrm{L}_2\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\). Then the distance between \(L_1\) and \(L_2\) is

87711 A vector a has components \(2 p\) and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If \(a\) has components \(p+1\) and 1 with respect to the new system, then

87708 Let \(L\) be a line passing through a point \(A\) and parallel to the vector \(2 \hat{i}+\hat{j}-2 \hat{k}\). Let \(-7 \hat{i}-5 \hat{j}+11 \hat{k}\) be the position vector of a point \(P\) on \(L\) such that \(|A P|=12\). Then, the position vector of \(A\) can be

87709 Let the vectors \(A B=2 \hat{i}+2 \hat{j}+\hat{k}\) and \(A C=2 \hat{i}+4 \hat{j}+4 \hat{k}\) be two sides of a \(\triangle A B C\). If \(G\) is the centroid of \(\triangle A B C\), then \(\frac{27}{7}|\mathrm{AG}|^2+5=\)

87710 \(\mathrm{L}_1\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}-\mathbf{3} \hat{\mathbf{k}}\). \(\mathrm{L}_2\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\). Then the distance between \(L_1\) and \(L_2\) is

87711 A vector a has components \(2 p\) and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If \(a\) has components \(p+1\) and 1 with respect to the new system, then

87708 Let \(L\) be a line passing through a point \(A\) and parallel to the vector \(2 \hat{i}+\hat{j}-2 \hat{k}\). Let \(-7 \hat{i}-5 \hat{j}+11 \hat{k}\) be the position vector of a point \(P\) on \(L\) such that \(|A P|=12\). Then, the position vector of \(A\) can be

87709 Let the vectors \(A B=2 \hat{i}+2 \hat{j}+\hat{k}\) and \(A C=2 \hat{i}+4 \hat{j}+4 \hat{k}\) be two sides of a \(\triangle A B C\). If \(G\) is the centroid of \(\triangle A B C\), then \(\frac{27}{7}|\mathrm{AG}|^2+5=\)

87710 \(\mathrm{L}_1\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}-\mathbf{3} \hat{\mathbf{k}}\). \(\mathrm{L}_2\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\). Then the distance between \(L_1\) and \(L_2\) is

87711 A vector a has components \(2 p\) and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If \(a\) has components \(p+1\) and 1 with respect to the new system, then