Angle Between Two Lines, Two Planes, a Line and a Plane
Three Dimensional Geometry

121308 The angle between the line \(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{k})\) and the plane \(\overline{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\mathbf{4}\)

1 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
3 \(\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
Three Dimensional Geometry

121310 The angle between the line \(\overline{\mathbf{r}}=(\mathbf{i}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \quad\) and the plane \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{3} \hat{\mathbf{k}})=\mathbf{8}\) is

1 \(\sin ^{-1}\left(\frac{2 \sqrt{7}}{\sqrt{5}}\right)\)
2 \(\sin ^{-1}\left(\frac{3 \sqrt{7}}{\sqrt{5}}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{7}}{3 \sqrt{5}}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\)
Three Dimensional Geometry

121311 The angle between the lines
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{k}}))+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}\) is

1 \(\cos ^{-1}\left(\frac{1}{5}\right)\)
2 \(\cos ^{-1}\left(\frac{1}{6}\right)\)
3 \(\cos ^{-1}\left(\frac{2}{3}\right)\)
4 \(\cos ^{-1}\left(\frac{1}{3}\right)\)
Three Dimensional Geometry

121320 The angle between lines \(\frac{x-2}{2}=\frac{y-3}{-2}=\frac{z-5}{1}\) and \(\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-5}{2}\) is

1 \(90^{\circ}\)
2 \(45^{\circ}\)
3 \(30^{\circ}\)
4 \(60^{\circ}\)
Three Dimensional Geometry

121308 The angle between the line \(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{k})\) and the plane \(\overline{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\mathbf{4}\)

1 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
3 \(\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
Three Dimensional Geometry

121310 The angle between the line \(\overline{\mathbf{r}}=(\mathbf{i}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \quad\) and the plane \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{3} \hat{\mathbf{k}})=\mathbf{8}\) is

1 \(\sin ^{-1}\left(\frac{2 \sqrt{7}}{\sqrt{5}}\right)\)
2 \(\sin ^{-1}\left(\frac{3 \sqrt{7}}{\sqrt{5}}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{7}}{3 \sqrt{5}}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\)
Three Dimensional Geometry

121311 The angle between the lines
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{k}}))+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}\) is

1 \(\cos ^{-1}\left(\frac{1}{5}\right)\)
2 \(\cos ^{-1}\left(\frac{1}{6}\right)\)
3 \(\cos ^{-1}\left(\frac{2}{3}\right)\)
4 \(\cos ^{-1}\left(\frac{1}{3}\right)\)
Three Dimensional Geometry

121320 The angle between lines \(\frac{x-2}{2}=\frac{y-3}{-2}=\frac{z-5}{1}\) and \(\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-5}{2}\) is

1 \(90^{\circ}\)
2 \(45^{\circ}\)
3 \(30^{\circ}\)
4 \(60^{\circ}\)
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Three Dimensional Geometry

121308 The angle between the line \(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{k})\) and the plane \(\overline{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\mathbf{4}\)

1 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
3 \(\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
Three Dimensional Geometry

121310 The angle between the line \(\overline{\mathbf{r}}=(\mathbf{i}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \quad\) and the plane \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{3} \hat{\mathbf{k}})=\mathbf{8}\) is

1 \(\sin ^{-1}\left(\frac{2 \sqrt{7}}{\sqrt{5}}\right)\)
2 \(\sin ^{-1}\left(\frac{3 \sqrt{7}}{\sqrt{5}}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{7}}{3 \sqrt{5}}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\)
Three Dimensional Geometry

121311 The angle between the lines
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{k}}))+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}\) is

1 \(\cos ^{-1}\left(\frac{1}{5}\right)\)
2 \(\cos ^{-1}\left(\frac{1}{6}\right)\)
3 \(\cos ^{-1}\left(\frac{2}{3}\right)\)
4 \(\cos ^{-1}\left(\frac{1}{3}\right)\)
Three Dimensional Geometry

121320 The angle between lines \(\frac{x-2}{2}=\frac{y-3}{-2}=\frac{z-5}{1}\) and \(\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-5}{2}\) is

1 \(90^{\circ}\)
2 \(45^{\circ}\)
3 \(30^{\circ}\)
4 \(60^{\circ}\)
Three Dimensional Geometry

121308 The angle between the line \(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{k})\) and the plane \(\overline{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\mathbf{4}\)

1 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
3 \(\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
Three Dimensional Geometry

121310 The angle between the line \(\overline{\mathbf{r}}=(\mathbf{i}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \quad\) and the plane \(\overline{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{3} \hat{\mathbf{k}})=\mathbf{8}\) is

1 \(\sin ^{-1}\left(\frac{2 \sqrt{7}}{\sqrt{5}}\right)\)
2 \(\sin ^{-1}\left(\frac{3 \sqrt{7}}{\sqrt{5}}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{7}}{3 \sqrt{5}}\right)\)
4 \(\sin ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\)
Three Dimensional Geometry

121311 The angle between the lines
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and
\(\overline{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{k}}))+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}\) is

1 \(\cos ^{-1}\left(\frac{1}{5}\right)\)
2 \(\cos ^{-1}\left(\frac{1}{6}\right)\)
3 \(\cos ^{-1}\left(\frac{2}{3}\right)\)
4 \(\cos ^{-1}\left(\frac{1}{3}\right)\)
Three Dimensional Geometry

121320 The angle between lines \(\frac{x-2}{2}=\frac{y-3}{-2}=\frac{z-5}{1}\) and \(\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-5}{2}\) is

1 \(90^{\circ}\)
2 \(45^{\circ}\)
3 \(30^{\circ}\)
4 \(60^{\circ}\)