Cartesian and Vector Equations of Line and Plane
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Three Dimensional Geometry

121188 The cosine of the angle included between the lines \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})\) and \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\) where \(\lambda, \mu \in R\) is

1 \(\frac{11}{21}\)
2 \(\frac{13}{21}\)
3 \(\frac{3}{21}\)
4 \(\frac{17}{21}\)
MHT CET-2020
Three Dimensional Geometry

121189 If the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\mathbf{2} \mathbf{j}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(\overrightarrow{\mathbf{r}} \cdot(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\mathbf{m} \hat{\mathbf{k}})=10\) then the value of \(m\) is

1 3
2 -2
3 -3
4 2
MHT CET-2020
Three Dimensional Geometry

121190 The vector equation of the line \(\frac{x+3}{2}=\frac{2 y-3}{5} ; z=-1\) is

1 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
2 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda\left(4 \hat{\mathrm{i}}+\frac{5}{2} \hat{\mathrm{j}}\right)\)
3 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}-\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
4 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}+\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
MHT CET-2020
Three Dimensional Geometry

121192 The position vector of the point of intersection of the line \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})\) and XOY-plane is

1 \(4 \hat{i}+3 \hat{k}\)
2 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{k}}\)
3 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\)
4 \(4 \hat{i}+3 \hat{j}\)
MHT CET-2020
Three Dimensional Geometry

121193 The point \(P\) lies on the line \(A B\), where \(A=(2,4,5) \quad\) and \(\quad B=(\mathbf{1}, 2,3) . \quad\) If \(\quad z\) coordinate of point \(P\) is 3 , then its \(y\) coordinate is

1 -2
2 -3
3 2
4 3
MHT CET-2020
NEET DIGITAL LIBRARY
Three Dimensional Geometry

121188 The cosine of the angle included between the lines \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})\) and \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\) where \(\lambda, \mu \in R\) is

1 \(\frac{11}{21}\)
2 \(\frac{13}{21}\)
3 \(\frac{3}{21}\)
4 \(\frac{17}{21}\)
MHT CET-2020
Three Dimensional Geometry

121189 If the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\mathbf{2} \mathbf{j}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(\overrightarrow{\mathbf{r}} \cdot(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\mathbf{m} \hat{\mathbf{k}})=10\) then the value of \(m\) is

1 3
2 -2
3 -3
4 2
MHT CET-2020
Three Dimensional Geometry

121190 The vector equation of the line \(\frac{x+3}{2}=\frac{2 y-3}{5} ; z=-1\) is

1 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
2 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda\left(4 \hat{\mathrm{i}}+\frac{5}{2} \hat{\mathrm{j}}\right)\)
3 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}-\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
4 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}+\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
MHT CET-2020
Three Dimensional Geometry

121192 The position vector of the point of intersection of the line \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})\) and XOY-plane is

1 \(4 \hat{i}+3 \hat{k}\)
2 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{k}}\)
3 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\)
4 \(4 \hat{i}+3 \hat{j}\)
MHT CET-2020
Three Dimensional Geometry

121193 The point \(P\) lies on the line \(A B\), where \(A=(2,4,5) \quad\) and \(\quad B=(\mathbf{1}, 2,3) . \quad\) If \(\quad z\) coordinate of point \(P\) is 3 , then its \(y\) coordinate is

1 -2
2 -3
3 2
4 3
MHT CET-2020
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Three Dimensional Geometry

121188 The cosine of the angle included between the lines \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})\) and \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\) where \(\lambda, \mu \in R\) is

1 \(\frac{11}{21}\)
2 \(\frac{13}{21}\)
3 \(\frac{3}{21}\)
4 \(\frac{17}{21}\)
MHT CET-2020
Three Dimensional Geometry

121189 If the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\mathbf{2} \mathbf{j}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(\overrightarrow{\mathbf{r}} \cdot(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\mathbf{m} \hat{\mathbf{k}})=10\) then the value of \(m\) is

1 3
2 -2
3 -3
4 2
MHT CET-2020
Three Dimensional Geometry

121190 The vector equation of the line \(\frac{x+3}{2}=\frac{2 y-3}{5} ; z=-1\) is

1 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
2 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda\left(4 \hat{\mathrm{i}}+\frac{5}{2} \hat{\mathrm{j}}\right)\)
3 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}-\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
4 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}+\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
MHT CET-2020
Three Dimensional Geometry

121192 The position vector of the point of intersection of the line \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})\) and XOY-plane is

1 \(4 \hat{i}+3 \hat{k}\)
2 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{k}}\)
3 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\)
4 \(4 \hat{i}+3 \hat{j}\)
MHT CET-2020
Three Dimensional Geometry

121193 The point \(P\) lies on the line \(A B\), where \(A=(2,4,5) \quad\) and \(\quad B=(\mathbf{1}, 2,3) . \quad\) If \(\quad z\) coordinate of point \(P\) is 3 , then its \(y\) coordinate is

1 -2
2 -3
3 2
4 3
MHT CET-2020
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Three Dimensional Geometry

121188 The cosine of the angle included between the lines \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})\) and \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\) where \(\lambda, \mu \in R\) is

1 \(\frac{11}{21}\)
2 \(\frac{13}{21}\)
3 \(\frac{3}{21}\)
4 \(\frac{17}{21}\)
MHT CET-2020
Three Dimensional Geometry

121189 If the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\mathbf{2} \mathbf{j}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(\overrightarrow{\mathbf{r}} \cdot(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\mathbf{m} \hat{\mathbf{k}})=10\) then the value of \(m\) is

1 3
2 -2
3 -3
4 2
MHT CET-2020
Three Dimensional Geometry

121190 The vector equation of the line \(\frac{x+3}{2}=\frac{2 y-3}{5} ; z=-1\) is

1 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
2 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda\left(4 \hat{\mathrm{i}}+\frac{5}{2} \hat{\mathrm{j}}\right)\)
3 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}-\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
4 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}+\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
MHT CET-2020
Three Dimensional Geometry

121192 The position vector of the point of intersection of the line \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})\) and XOY-plane is

1 \(4 \hat{i}+3 \hat{k}\)
2 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{k}}\)
3 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\)
4 \(4 \hat{i}+3 \hat{j}\)
MHT CET-2020
Three Dimensional Geometry

121193 The point \(P\) lies on the line \(A B\), where \(A=(2,4,5) \quad\) and \(\quad B=(\mathbf{1}, 2,3) . \quad\) If \(\quad z\) coordinate of point \(P\) is 3 , then its \(y\) coordinate is

1 -2
2 -3
3 2
4 3
MHT CET-2020
NEET DIGITAL LIBRARY
Three Dimensional Geometry

121188 The cosine of the angle included between the lines \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}})\) and \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\) where \(\lambda, \mu \in R\) is

1 \(\frac{11}{21}\)
2 \(\frac{13}{21}\)
3 \(\frac{3}{21}\)
4 \(\frac{17}{21}\)
MHT CET-2020
Three Dimensional Geometry

121189 If the line \(\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\mathbf{2} \mathbf{j}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(\overrightarrow{\mathbf{r}} \cdot(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\mathbf{m} \hat{\mathbf{k}})=10\) then the value of \(m\) is

1 3
2 -2
3 -3
4 2
MHT CET-2020
Three Dimensional Geometry

121190 The vector equation of the line \(\frac{x+3}{2}=\frac{2 y-3}{5} ; z=-1\) is

1 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
2 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda\left(4 \hat{\mathrm{i}}+\frac{5}{2} \hat{\mathrm{j}}\right)\)
3 \(\overrightarrow{\mathrm{r}}=\left(3 \hat{\mathrm{i}}-\frac{3}{2} \hat{\mathrm{j}}-\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
4 \(\overrightarrow{\mathrm{r}}=\left(-3 \hat{\mathrm{i}}+\frac{3}{2} \hat{\mathrm{j}}+\hat{\mathrm{k}}\right)+\lambda(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})\)
MHT CET-2020
Three Dimensional Geometry

121192 The position vector of the point of intersection of the line \(\overrightarrow{\mathbf{r}}=(\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}})\) and XOY-plane is

1 \(4 \hat{i}+3 \hat{k}\)
2 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{k}}\)
3 \(4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\)
4 \(4 \hat{i}+3 \hat{j}\)
MHT CET-2020
Three Dimensional Geometry

121193 The point \(P\) lies on the line \(A B\), where \(A=(2,4,5) \quad\) and \(\quad B=(\mathbf{1}, 2,3) . \quad\) If \(\quad z\) coordinate of point \(P\) is 3 , then its \(y\) coordinate is

1 -2
2 -3
3 2
4 3
MHT CET-2020