121321 If the angle between the planes
\(\overline{\mathbf{r}} \times(\mathbf{m} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\mathbf{3}=\mathbf{0}\) and
\(\overline{\mathbf{r}} \times(2 \hat{\mathbf{i}}-\mathbf{m} \hat{\mathbf{j}}-\hat{\mathbf{k}})-\mathbf{5}=\mathbf{0}\) is \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=6\), then \(\mathbf{m}=\)

121328 Let the points on the plane \(P\) be equidistant from the points \((-4,2,1)\) and \((2,-2,3)\). Then the acute angle between the plane \(P\) and the plane \(2 x+y+3 z=1\) is

121330 A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}\). The obtuse angle between \(\vec{a}\) and the vector \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) is

121332 The angle between the lines \(\hat{\mathbf{r}}=(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})\) and \(\hat{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})\) is

121321 If the angle between the planes
\(\overline{\mathbf{r}} \times(\mathbf{m} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\mathbf{3}=\mathbf{0}\) and
\(\overline{\mathbf{r}} \times(2 \hat{\mathbf{i}}-\mathbf{m} \hat{\mathbf{j}}-\hat{\mathbf{k}})-\mathbf{5}=\mathbf{0}\) is \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=6\), then \(\mathbf{m}=\)

121328 Let the points on the plane \(P\) be equidistant from the points \((-4,2,1)\) and \((2,-2,3)\). Then the acute angle between the plane \(P\) and the plane \(2 x+y+3 z=1\) is

121330 A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}\). The obtuse angle between \(\vec{a}\) and the vector \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) is

121332 The angle between the lines \(\hat{\mathbf{r}}=(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})\) and \(\hat{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})\) is

121321 If the angle between the planes
\(\overline{\mathbf{r}} \times(\mathbf{m} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\mathbf{3}=\mathbf{0}\) and
\(\overline{\mathbf{r}} \times(2 \hat{\mathbf{i}}-\mathbf{m} \hat{\mathbf{j}}-\hat{\mathbf{k}})-\mathbf{5}=\mathbf{0}\) is \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=6\), then \(\mathbf{m}=\)

121328 Let the points on the plane \(P\) be equidistant from the points \((-4,2,1)\) and \((2,-2,3)\). Then the acute angle between the plane \(P\) and the plane \(2 x+y+3 z=1\) is

121330 A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}\). The obtuse angle between \(\vec{a}\) and the vector \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) is

121332 The angle between the lines \(\hat{\mathbf{r}}=(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})\) and \(\hat{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})\) is

121321 If the angle between the planes
\(\overline{\mathbf{r}} \times(\mathbf{m} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})+\mathbf{3}=\mathbf{0}\) and
\(\overline{\mathbf{r}} \times(2 \hat{\mathbf{i}}-\mathbf{m} \hat{\mathbf{j}}-\hat{\mathbf{k}})-\mathbf{5}=\mathbf{0}\) is \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=6\), then \(\mathbf{m}=\)

121328 Let the points on the plane \(P\) be equidistant from the points \((-4,2,1)\) and \((2,-2,3)\). Then the acute angle between the plane \(P\) and the plane \(2 x+y+3 z=1\) is

121330 A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}\). The obtuse angle between \(\vec{a}\) and the vector \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) is

121332 The angle between the lines \(\hat{\mathbf{r}}=(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})\) and \(\hat{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})\) is