121233 Let \(\mathrm{m}\) be the unit vector orthogonal to the vector \(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and coplanar with the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\). If \(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}\), then the length of the perpendicular from the origin to the plane \(\mathbf{r} \cdot \mathbf{m}=\mathbf{a} \cdot \mathbf{m}\) is

121235 The shortest distance between the lines \(\mathbf{r}=(3 \mathrm{i}+\) \(5 \mathbf{j}+7 \mathbf{k})+\lambda(\mathbf{i}+2 \mathbf{j}+\mathbf{k})\) and \(\mathbf{r}=(-\mathbf{i}-\mathbf{j}-\mathbf{k})+\mu\) \((7 \mathbf{i}-\mathbf{j} \mathbf{j} \mathbf{k})\) is

121219 The shortest distance between the line \(1+x=2 y=-12 z\) and \(x=y+2=6 z-6\) is

121228 The shortest distance between the lines \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}\) and \(\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}\)

121233 Let \(\mathrm{m}\) be the unit vector orthogonal to the vector \(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and coplanar with the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\). If \(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}\), then the length of the perpendicular from the origin to the plane \(\mathbf{r} \cdot \mathbf{m}=\mathbf{a} \cdot \mathbf{m}\) is

121235 The shortest distance between the lines \(\mathbf{r}=(3 \mathrm{i}+\) \(5 \mathbf{j}+7 \mathbf{k})+\lambda(\mathbf{i}+2 \mathbf{j}+\mathbf{k})\) and \(\mathbf{r}=(-\mathbf{i}-\mathbf{j}-\mathbf{k})+\mu\) \((7 \mathbf{i}-\mathbf{j} \mathbf{j} \mathbf{k})\) is

121219 The shortest distance between the line \(1+x=2 y=-12 z\) and \(x=y+2=6 z-6\) is

121228 The shortest distance between the lines \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}\) and \(\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}\)

121233 Let \(\mathrm{m}\) be the unit vector orthogonal to the vector \(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and coplanar with the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\). If \(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}\), then the length of the perpendicular from the origin to the plane \(\mathbf{r} \cdot \mathbf{m}=\mathbf{a} \cdot \mathbf{m}\) is

121235 The shortest distance between the lines \(\mathbf{r}=(3 \mathrm{i}+\) \(5 \mathbf{j}+7 \mathbf{k})+\lambda(\mathbf{i}+2 \mathbf{j}+\mathbf{k})\) and \(\mathbf{r}=(-\mathbf{i}-\mathbf{j}-\mathbf{k})+\mu\) \((7 \mathbf{i}-\mathbf{j} \mathbf{j} \mathbf{k})\) is

121219 The shortest distance between the line \(1+x=2 y=-12 z\) and \(x=y+2=6 z-6\) is

121228 The shortest distance between the lines \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}\) and \(\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}\)

121233 Let \(\mathrm{m}\) be the unit vector orthogonal to the vector \(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and coplanar with the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\). If \(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}\), then the length of the perpendicular from the origin to the plane \(\mathbf{r} \cdot \mathbf{m}=\mathbf{a} \cdot \mathbf{m}\) is

121235 The shortest distance between the lines \(\mathbf{r}=(3 \mathrm{i}+\) \(5 \mathbf{j}+7 \mathbf{k})+\lambda(\mathbf{i}+2 \mathbf{j}+\mathbf{k})\) and \(\mathbf{r}=(-\mathbf{i}-\mathbf{j}-\mathbf{k})+\mu\) \((7 \mathbf{i}-\mathbf{j} \mathbf{j} \mathbf{k})\) is

121219 The shortest distance between the line \(1+x=2 y=-12 z\) and \(x=y+2=6 z-6\) is

121228 The shortest distance between the lines \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}\) and \(\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}\)