Cartesian and Vector Equations of Line and Plane
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Three Dimensional Geometry

121205 The equation of the plane through the intersection of the planes \(3 x-y+2 z-4=0\) and \(x+y+z-2=0\) and the point \((2,2,1)\) is

1 \(7 x+5 y+4 z+8=0\)
2 \(7 x+5 y+4 z-8=0\)
3 \(7 x-5 y+4 z-8=0\)
4 None of these
Manipal UGET-2014
Three Dimensional Geometry

121207 The equation of plane passing through a point A \((2,-1,3)\) and parallel to the vectors \(\vec{a}=(3,0\), \(-1)\) and \(\vec{b}=(-3,2,2)\) is

1 \(2 x-3 y+6 z+25=0\)
2 \(3 x-2 y+6 z+25=0\)
3 \(2 x-3 y+6 z-25=0\)
4 \(3 x-2 y+6 z-25=0\)
Manipal UGET-2011
Three Dimensional Geometry

121209 Equation of a plane passing through \((-1,1,1)\) and \((1,-1,1)\) and perpendicular to \(x+2 y+2 z\) \(=\mathbf{5}\) is

1 \(2 x+3 y-3 z+3=0\)
2 \(x+y+3 z-5=0\)
3 \(2 x+2 y-3 z+3=0\)
4 \(x+y+z-3=0\)
Manipal UGET-2020
Three Dimensional Geometry

121210 The shortest distance between the line \(\mathbf{r}=\mathbf{2} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}})\) and the plane r. \((\hat{\mathbf{i}}+\mathbf{5} \hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\frac{1}{3 \sqrt{3}}\)
2 \(\frac{5}{3 \sqrt{3}}\)
3 \(\frac{10}{3 \sqrt{3}}\)
4 \(\frac{11}{3 \sqrt{3}}\)
TS EAMCET-10.09.2020
Three Dimensional Geometry

121191 The acute angle between the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})\) and the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\sin ^{-1}\left(\sqrt{\frac{2}{3}}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
MHT CET-2020
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Three Dimensional Geometry

121205 The equation of the plane through the intersection of the planes \(3 x-y+2 z-4=0\) and \(x+y+z-2=0\) and the point \((2,2,1)\) is

1 \(7 x+5 y+4 z+8=0\)
2 \(7 x+5 y+4 z-8=0\)
3 \(7 x-5 y+4 z-8=0\)
4 None of these
Manipal UGET-2014
Three Dimensional Geometry

121207 The equation of plane passing through a point A \((2,-1,3)\) and parallel to the vectors \(\vec{a}=(3,0\), \(-1)\) and \(\vec{b}=(-3,2,2)\) is

1 \(2 x-3 y+6 z+25=0\)
2 \(3 x-2 y+6 z+25=0\)
3 \(2 x-3 y+6 z-25=0\)
4 \(3 x-2 y+6 z-25=0\)
Manipal UGET-2011
Three Dimensional Geometry

121209 Equation of a plane passing through \((-1,1,1)\) and \((1,-1,1)\) and perpendicular to \(x+2 y+2 z\) \(=\mathbf{5}\) is

1 \(2 x+3 y-3 z+3=0\)
2 \(x+y+3 z-5=0\)
3 \(2 x+2 y-3 z+3=0\)
4 \(x+y+z-3=0\)
Manipal UGET-2020
Three Dimensional Geometry

121210 The shortest distance between the line \(\mathbf{r}=\mathbf{2} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}})\) and the plane r. \((\hat{\mathbf{i}}+\mathbf{5} \hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\frac{1}{3 \sqrt{3}}\)
2 \(\frac{5}{3 \sqrt{3}}\)
3 \(\frac{10}{3 \sqrt{3}}\)
4 \(\frac{11}{3 \sqrt{3}}\)
TS EAMCET-10.09.2020
Three Dimensional Geometry

121191 The acute angle between the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})\) and the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\sin ^{-1}\left(\sqrt{\frac{2}{3}}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
MHT CET-2020
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Three Dimensional Geometry

121205 The equation of the plane through the intersection of the planes \(3 x-y+2 z-4=0\) and \(x+y+z-2=0\) and the point \((2,2,1)\) is

1 \(7 x+5 y+4 z+8=0\)
2 \(7 x+5 y+4 z-8=0\)
3 \(7 x-5 y+4 z-8=0\)
4 None of these
Manipal UGET-2014
Three Dimensional Geometry

121207 The equation of plane passing through a point A \((2,-1,3)\) and parallel to the vectors \(\vec{a}=(3,0\), \(-1)\) and \(\vec{b}=(-3,2,2)\) is

1 \(2 x-3 y+6 z+25=0\)
2 \(3 x-2 y+6 z+25=0\)
3 \(2 x-3 y+6 z-25=0\)
4 \(3 x-2 y+6 z-25=0\)
Manipal UGET-2011
Three Dimensional Geometry

121209 Equation of a plane passing through \((-1,1,1)\) and \((1,-1,1)\) and perpendicular to \(x+2 y+2 z\) \(=\mathbf{5}\) is

1 \(2 x+3 y-3 z+3=0\)
2 \(x+y+3 z-5=0\)
3 \(2 x+2 y-3 z+3=0\)
4 \(x+y+z-3=0\)
Manipal UGET-2020
Three Dimensional Geometry

121210 The shortest distance between the line \(\mathbf{r}=\mathbf{2} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}})\) and the plane r. \((\hat{\mathbf{i}}+\mathbf{5} \hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\frac{1}{3 \sqrt{3}}\)
2 \(\frac{5}{3 \sqrt{3}}\)
3 \(\frac{10}{3 \sqrt{3}}\)
4 \(\frac{11}{3 \sqrt{3}}\)
TS EAMCET-10.09.2020
Three Dimensional Geometry

121191 The acute angle between the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})\) and the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\sin ^{-1}\left(\sqrt{\frac{2}{3}}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
MHT CET-2020
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Three Dimensional Geometry

121205 The equation of the plane through the intersection of the planes \(3 x-y+2 z-4=0\) and \(x+y+z-2=0\) and the point \((2,2,1)\) is

1 \(7 x+5 y+4 z+8=0\)
2 \(7 x+5 y+4 z-8=0\)
3 \(7 x-5 y+4 z-8=0\)
4 None of these
Manipal UGET-2014
Three Dimensional Geometry

121207 The equation of plane passing through a point A \((2,-1,3)\) and parallel to the vectors \(\vec{a}=(3,0\), \(-1)\) and \(\vec{b}=(-3,2,2)\) is

1 \(2 x-3 y+6 z+25=0\)
2 \(3 x-2 y+6 z+25=0\)
3 \(2 x-3 y+6 z-25=0\)
4 \(3 x-2 y+6 z-25=0\)
Manipal UGET-2011
Three Dimensional Geometry

121209 Equation of a plane passing through \((-1,1,1)\) and \((1,-1,1)\) and perpendicular to \(x+2 y+2 z\) \(=\mathbf{5}\) is

1 \(2 x+3 y-3 z+3=0\)
2 \(x+y+3 z-5=0\)
3 \(2 x+2 y-3 z+3=0\)
4 \(x+y+z-3=0\)
Manipal UGET-2020
Three Dimensional Geometry

121210 The shortest distance between the line \(\mathbf{r}=\mathbf{2} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}})\) and the plane r. \((\hat{\mathbf{i}}+\mathbf{5} \hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\frac{1}{3 \sqrt{3}}\)
2 \(\frac{5}{3 \sqrt{3}}\)
3 \(\frac{10}{3 \sqrt{3}}\)
4 \(\frac{11}{3 \sqrt{3}}\)
TS EAMCET-10.09.2020
Three Dimensional Geometry

121191 The acute angle between the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})\) and the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\sin ^{-1}\left(\sqrt{\frac{2}{3}}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
MHT CET-2020
Join With Us for More Updates
Three Dimensional Geometry

121205 The equation of the plane through the intersection of the planes \(3 x-y+2 z-4=0\) and \(x+y+z-2=0\) and the point \((2,2,1)\) is

1 \(7 x+5 y+4 z+8=0\)
2 \(7 x+5 y+4 z-8=0\)
3 \(7 x-5 y+4 z-8=0\)
4 None of these
Manipal UGET-2014
Three Dimensional Geometry

121207 The equation of plane passing through a point A \((2,-1,3)\) and parallel to the vectors \(\vec{a}=(3,0\), \(-1)\) and \(\vec{b}=(-3,2,2)\) is

1 \(2 x-3 y+6 z+25=0\)
2 \(3 x-2 y+6 z+25=0\)
3 \(2 x-3 y+6 z-25=0\)
4 \(3 x-2 y+6 z-25=0\)
Manipal UGET-2011
Three Dimensional Geometry

121209 Equation of a plane passing through \((-1,1,1)\) and \((1,-1,1)\) and perpendicular to \(x+2 y+2 z\) \(=\mathbf{5}\) is

1 \(2 x+3 y-3 z+3=0\)
2 \(x+y+3 z-5=0\)
3 \(2 x+2 y-3 z+3=0\)
4 \(x+y+z-3=0\)
Manipal UGET-2020
Three Dimensional Geometry

121210 The shortest distance between the line \(\mathbf{r}=\mathbf{2} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{4} \hat{\mathbf{k}})\) and the plane r. \((\hat{\mathbf{i}}+\mathbf{5} \hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\frac{1}{3 \sqrt{3}}\)
2 \(\frac{5}{3 \sqrt{3}}\)
3 \(\frac{10}{3 \sqrt{3}}\)
4 \(\frac{11}{3 \sqrt{3}}\)
TS EAMCET-10.09.2020
Three Dimensional Geometry

121191 The acute angle between the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})\) and the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5\) is

1 \(\sin ^{-1}\left(\sqrt{\frac{2}{3}}\right)\)
2 \(\sin ^{-1}\left(\frac{2}{3}\right)\)
3 \(\sin ^{-1}\left(\frac{\sqrt{2}}{3}\right)\)
4 \(\sin ^{-1}\left(\frac{2}{\sqrt{3}}\right)\)
MHT CET-2020