Find the equation of the plane passing through the points
Question: Find the equation of the plane passing through the points $A(1,1,2)$ and $B(2,-2,2)$ and perpendicular to the plane $6 x-2 y+2 z=9$. Solution:...
Read More →Find the equation of the plane that contains the point
Question: Find the equation of the plane that contains the point $A(1,-1,2)$ and is perpendicular to both the planes $3 x+3 y-2 z=5$ and $x+2 y-3 z=$ 8. Hence, find the distance of the point $P(-2,5,5)$ from the plane obtained above. Solution:...
Read More →Find the equation of the plane passing through the origin
Question: Find the equation of the plane passing through the origin and perpendicular to each of the planes $x+2 y-z=1$ and $3 x-4 y+z=5$. Solution:...
Read More →Find the equation of the plane passing through the point
Question: Find the equation of the plane passing through the point $A(-1,-1,2)$ and perpendicular to each of the planes $3 x+2 y-3 z=1$ and $5 x-4 y+$ $z=5$. Solution:...
Read More →Find the equation of the plane passing through the point
Question: Find the equation of the plane passing through the point $(1,-2,7)$ and parallel to the plane $5 x+4 y-11 z=6$. Solution:...
Read More →Find the equation of the plane passing through the point
Question: Find the equation of the plane passing through the point $(a, b, c)$ and parallel to the plane $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=2$. Solution:...
Read More →Write the equation of the plane passing through the origin and parallel to the plane
Question: Write the equation of the plane passing through the origin and parallel to the plane $5 x-3 y+7 z+11=0$. Solution:...
Read More →Solve this following
Question: Find the value of $\lambda$ for which the planes $x-4 y+\lambda z+3=0$ and $2 x+2 y+3 z=5$ are perpendicular to each other. Solution:...
Read More →Solve this following
Question: Show that the planes $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$ are parallel. Solution:...
Read More →Prove that the plane
Question: Prove that the plane $2 x+2 y+4 z=9$ is perpendicular to each of the planes $x+2 y+2 z-7=0$ and $5 x+6 y+7 z=23$. Solution:...
Read More →Show that each of the following pairs of planes are at right angles:
Question: Show that each of the following pairs of planes are at right angles: (i) $3 x+4 y-5 z=7$ and $2 x+6 y+6 z+7=0$ (ii) $x-2 y+4 z=10$ and $18 x+17 y+4 z=49$ Solution:...
Read More →Find the acute angle between the following planes:
Question: Find the acute angle between the following planes: (i) $2 x-y+z=5$ and $x+y+2 z=7$ (ii) $x+2 y+2 z=3$ and $2 x-3 y+6 z=8$ (iii) $x+y-z=4$ and $x+2 y+z=9$ (iv) $x+y-2 z=6$ and $2 x-2 y+z=11$ Solution:...
Read More →Solve this following
Question: Find the value of $\lambda$ for which the given planes are perpendicular to each other: (i) $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}})=7$ and $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=9$ (ii) $\overrightarrow{\mathrm{r}} \cdot(\lambda \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=5$ and $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-7 \hat{\mathrm{k...
Read More →Show that the following planes are at right angles:
Question: Show that the following planes are at right angles: (i) $\overrightarrow{\mathrm{r}} \cdot(4 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-8 \hat{\mathrm{k}})=5$ and $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})+10=0$ (ii) $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})=13$ and $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+7=0$ Solution:...
Read More →Find the acute angle between the following planes:
Question: Find the acute angle between the following planes: (i) $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})=5$ and $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})=9$ (ii) $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})=6$ and $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})+3=0$ (iii) $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\m...
Read More →Find the Cartesian and vector equations of the planes through the line of intersection of the planes
Question: Find the Cartesian and vector equations of the planes through the line of intersection of the planes $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{j}})+6=0$ and $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})=0$, which are at a unit distance from the origin. Solution:...
Read More →Find the equation of the plane through the line of intersection of the planes
Question: Find the equation of the plane through the line of intersection of the planes $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=1$ and $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{j}})+4=0$ and perpendicular to the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+8=0$ Solution:...
Read More →Find the vector equation of the plane passing through the intersection of the planes
Question: Find the vector equation of the plane passing through the intersection of the planes$\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=3$ and $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})+11=0$, and passing through the point $(-2,1,3)$. Solution:...
Read More →Find the vector equation of the plane through the point
Question: Find the vector equation of the plane through the point $(1,1,1)$, and passing through the intersection of the planes $\vec{r} \cdot(\hat{i}-\hat{j}+3 \hat{k})+1=0$ and $\overrightarrow{\mathrm{r}} .(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})-5=0$ Solution:...
Read More →Find the vector equation of the plane through the intersection of the planes
Question: Find the vector equation of the plane through the intersection of the planes$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})=0$ and $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{j}}+2 \hat{\mathrm{k}})=0$. and passing through the point $(2,1,-1)$ Solution:...
Read More →Find the equation of the plane through the intersection of the planes
Question: Find the equation of the plane through the intersection of the planes $3 x-4 y+5 z=10$ and $2 x+2 y-3 z=4$ and parallel to the line $x=2 y=$ $3 z$. Solution:...
Read More →Find the equation of the plane passing through the line of intersection
Question: Find the equation of the plane passing through the line of intersection of the planes $x+2 y+3 z-5=0$ and $3 x-2 y-z+1=0$ and cutting off equal intercepts on the $x$-axis and $z$-axis. Solution:...
Read More →Find the equation of the plane passing through the intersection of the planes
Question: Find the equation of the plane passing through the intersection of the planes $x-2 y+z=1$ and $2 x+y+z=8$, and parallel to the line with direction ratios $1,2,1$. Also, find the perpendicular distance of $(1,1,1)$ from the plane. Solution:...
Read More →Find the equation of the plane passing through the line of intersection of the planes
Question: Find the equation of the plane passing through the line of intersection of the planes $2 x-y=0$ and $3 z-y=0$, and perpendicular to the plane $4 x+5 y-3 z=9$ Solution:...
Read More →Find the equation of the plane passing through the intersection of the planes
Question: Find the equation of the plane passing through the intersection of the planes $2 x+3 y-z+1=0$ and $x+y-2 z+3=0$, and perpendicular to the plane $3 x-y-2 z-4=0$. Solution:...
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