Find the vector and Cartesian forms of the equations of the plane containing the two lines
Question: Find the vector and Cartesian forms of the equations of the plane containing the two lines $\bar{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and $\bar{r}=(9 \hat{i}+5 \hat{j}-\hat{k})+\mu(-2 \hat{i}+3 \hat{j}+8 \hat{k}) \ldots$ Solution:...
Read More →Show that the lines
Question: Show that the lines $\bar{r}=(2 \hat{j}-3 \hat{k})+\hat{\lambda}(\hat{j}+2 \hat{j}+3 \hat{k})$ and $\bar{r}=(2 \hat{i}+6 \hat{j}+3 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+4 \hat{k})$ are coplanar. Also find the equation of the plane containing these lines. Solution:...
Read More →Find the vector equation of the plane passing through the point
Question: Find the vector equation of the plane passing through the point $(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ and parallel to the vectors $(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ and $(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$. Solution:...
Read More →Find the Cartesian and vector equations of a plane passing through the point
Question: Find the Cartesian and vector equations of a plane passing through the point $(1,2,-4)$ and parallel to the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and $\frac{x-1}{1}=\frac{y+3}{1}=\frac{z}{-1}$ Solution:...
Read More →Find the vector equation of a plane passing through the point
Question: Find the vector equation of a plane passing through the point $(1,2,3)$ and parallel to the lines whose direction ratios are 1 , - 1 , - 2 , and - 1 , 0,2 . Solution:...
Read More →Find the vector and Cartesian equations of the plane passing
Question: Find the vector and Cartesian equations of the plane passing through the point $(3,-1,2)$ and parallel to the lines $\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(-5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}})$. Solution:...
Read More →Find the vector and Cartesian equations
Question: Find the vector and Cartesian equations of the plane passing through the origin and parallel to the vectors $(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $(3 \hat{\mathrm{i}}-\hat{\mathrm{k}})$. Solution:...
Read More →Find the equation of the plane passing through the intersection
Question: Find the equation of the plane passing through the intersection of the planes5x $-y+z=10$ and $x+y-z=4$ and parallel to the line with direction ratios2, 1,1 . Find also the perpendicular distance of $(1,1,1)$ from this plane. Solution:...
Read More →Find the equation of a plane passing through the point
Question: Find the equation of a plane passing through the point $(2,-1,5)$, perpendicular to the plane $x+2 y-3 z=7$ and parallel to the line $\frac{x+5}{3}=\frac{y+1}{-1}=\frac{z-2}{1}$ Solution:...
Read More →Find the equation of the plane passing through the points
Question: Find the equation of the plane passing through the points $(1,2,3)$ and $(0,-1,0)$ and parallel to the line $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z}{-3}$. Solution:...
Read More →Solve this following
Question: Show that the equation $a x+b y+d=0$ represents a plane parallel to the $z$ - axis. Hence, find the equation of a plane which is parallel to the $z$ - axis and passes through the points $A(2,-3,1)$ and $B(-4,7,6)$. Solution:...
Read More →Find the vector equation of the line passing through the point with position vector
Question: Find the vector equation of the line passing through the point with position vector $(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$ and perpendicular to the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}-\hat{\mathrm{k}})=0$ Solution:...
Read More →Find the vector equation of a line passing through the origin and perpendicular
Question: Find the vector equation of a line passing through the origin and perpendicular to the plane $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=3$. Solution:...
Read More →Solve this following
Question: Find the value of $m$ for which the line $\vec{r}=(\hat{i}+2 \hat{k})+\lambda(2 \hat{i}-m \hat{j}-3 \hat{k})$ is parallel to the plane $\vec{r} \cdot(m \hat{i}+3 \hat{j}+\hat{k})=4$. Solution:...
Read More →Solve this following
Question: Show that the line $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+7 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$ is parallel to the plane $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})=7$. Also, find the distance between them. Solution:...
Read More →Solve this following
Question: If the plane $2 x-3 y-6 z=13$ makes an angle $\sin ^{-1}(\lambda)$ with the $x$-axis, then find the value of $\lambda$. Solution:...
Read More →Find the angle between the line joining the points
Question: Find the angle between the line joining the points $A(3,-4,-2)$ and $B(12,2,0)$ and the plane $3 x-y+z=1$. Solution:...
Read More →Find the angle between the line
Question: Find the angle between the line $\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}$ and the plane $10 x+2 y-11 z=3$. Solution:...
Read More →Find the angle between the line
Question: Find the angle between the line $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{z-3}{2}$ and the plane $3 x+4 y+z+5=0$. Solution:...
Read More →Find the angle between the line
Question: Find the angle between the line $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{k}})+\lambda(\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})=1$. Solution:...
Read More →Find the angle between the line
Question: Find the angle between the line $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ and the plane $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=3$. Solution:...
Read More →Solve this following
Question: Find the angle between the line $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})=4$. Solution:...
Read More →Find the equation of the plane through the points
Question: Find the equation of the plane through the points $A(2,1,-1)$ and $B(-1,3,4)$ and perpendicular to the plane $x-2 y+4 z=10 .$ Also, show that the plane thus obtained contains the line $\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})+\lambda(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ Solution:...
Read More →Find the equation of the plane through the points
Question: Find the equation of the plane through the points $A(3,4,2)$ and $B(7,0,6)$ and perpendicular to the plane $2 x-5 y=15$. HINT: The given plane is $2 x-5 y+0 z=15$ Solution:...
Read More →Find the equation of the plane passing through the points
Question: Find the equation of the plane passing through the points $A(-1,1,1)$ and $B(1,-1,1)$ and perpendicular to the plane $x+2 y+2 z=5$. Solution:...
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