Solve this following
Question: A vector $\bar{n}$ of magnitude 8 units is inclined to the $x$-axis at $45^{\circ}, y$-axis at $60^{\circ}$ and an acute angle with the $z$-axis, if a plane passes through a point $(\sqrt{2},-1,1)$ and is normal to find its equation in vector form. Solution:...
Read More →Find the equation of a plane which is at a distance
Question: Find the equation of a plane which is at a distance of $3 \sqrt{3}$ units from the origin and the normal to which is equally inclined to the coordinate axes. Solution:...
Read More →Solve this following
Question: Show that the line $\vec{r}-(2 \hat{i}-2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-\hat{j}+4 \hat{k})$ is parallel to the plane $\vec{r} \cdot(\hat{i}+5 \hat{j}+\hat{k})-7$. Solution:...
Read More →Prove that the normals to the planes
Question: Prove that the normals to the planes $4 x+11 y+2 z+3=0$ and $3 x-2 y+5 z=8$ are perpendicular to each other. Solution:...
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Question: Find the distance of the point $\mathrm{A}(-1,-5,-10)$ from the point of intersection of the line $\vec{r}=(2 \hat{i}-\hat{j}+2 \hat{k})+\lambda(3 \hat{i}+4 \hat{j}+2 \hat{k})$ and the plane $\vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=5$. HINT: Convert the equations of the line and the plane to Cartesian form. Solution:...
Read More →Solve this following
Question: Show that the distance of the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane $x-y+z=5$ from the point $(-1,-5-10)$ is 13 units. Solution:...
Read More →Find the equation of the line passing through the point
Question: Find the equation of the line passing through the point $P(4,6,2)$ and the point of intersection of the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z+1}{7}$ and the plane $x+y-Z$ $=8$. Solution:...
Read More →Find the distance of the point
Question: Find the distance of the point $(0,-3,2)$ from the plane $x+2 y-z=1$, measured parallel to the line $\frac{x-1}{3}=\frac{y-1}{2}=\frac{z}{3}$. Solution:...
Read More →Find the distance of the point
Question: Find the distance of the point $(2,3,4)$ from the plane $3 x+2 y+2 z+5=0$, measured parallel to the line $\frac{x-3}{3}=\frac{y-2}{6}=\frac{z}{2}$. Solution:...
Read More →Find the coordinates of the point whepe the line through
Question: Find the coordinates of the point whepe the line through $(3,-4,-5)$ and $(2,-3,1)$ crossess the plane $2 x+y+z=7$. Solution:...
Read More →Find the point where the line
Question: Find the point where the line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+3}{4}$ meets the plane $2 x+4 y-z=1$. Solution:...
Read More →Find the coordinates of the image of the point
Question: Find the coordinates of the image of the point $P(1,3,4)$ in the plane $2 x-y+z+3=0$. Solution:...
Read More →Find the coordinates of the foot of the perpendicular and the perpendicular distance from the point
Question: Find the coordinates of the foot of the perpendicular and the perpendicular distance from the point $P(3,2,1)$ to the plane $2 x-y+z+1=$ 0 . Find also the image of the point P in the plane. Solution:...
Read More →Solve this following
Question: From the point $P(1,2,4)$, a perpendicular is drawn on the plane $2 x+y-2 z+3=0$. Find the equation, the length and the coordinates of the foot of the perpendicular. Solution:...
Read More →Find the length and the foot of the perpendicular drawn from the point
Question: Find the length and the foot of the perpendicular drawn from the point $(1,1,2)$ to the plane $\bar{r} \cdot(2 \hat{i}-2 \hat{j}+4 \hat{k})+5-0$. Solution:...
Read More →Find the length and the foot of perpendicular drawn from the point
Question: Find the length and the foot of perpendicular drawn from the point $(2,3,7)$ to the plane $3 x-y-z=7$. Solution:...
Read More →Find the coordinates of the foot of the perpendicular drawn from the origin to the plane
Question: Find the coordinates of the foot of the perpendicular drawn from the origin to the plane (i) $2 x+3 y+4 z-12=0$ (ii) $5 y+8=0$ Solution:...
Read More →Find the vector and Cartesian equations of the plane passing through the point
Question: Find the vector and Cartesian equations of the plane passing through the point $(2,-1,1)$ and perpendicular to the line having direction ratios $4,2,-3$. Solution:...
Read More →For each of the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin:
Question: For each of the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin: (i) $2 x+3 y-z=5$ (ii) $z=3$ (iii) $3 y+5=0$ Solution:...
Read More →Find the direction cosines of the normal to the plane
Question: Find the direction cosines of the normal to the plane $3 x-6 y+2 z=7$. Solution:...
Read More →Find a unit vector normal to the planex
Question: Find a unit vector normal to the planex $-2 y+2 z=6$ Solution:...
Read More →Find the vector equation of a plane whose Cartesian equation is
Question: Find the vector equation of a plane whose Cartesian equation is $5 x-7 y+2 z+4=0$. Solution:...
Read More →Find the Cartesian equation of the plane whose vector equation is
Question: Find the Cartesian equation of the plane whose vector equation is $\vec{r} \cdot(3 \hat{i}+5 \hat{j}-9 \hat{k})-8$. Solution:...
Read More →Find the length of the perpendicular from the origin to the plane
Question: Find the length of the perpendicular from the origin to the plane $\vec{r} \cdot(3 \hat{i}-12 \hat{j}-4 \hat{k})+39-0$. Also write the unit normal vector from the origin to the plane. Solution:...
Read More →Find the vector, and Cartesian equations of a plane which passes through the point
Question: Find the vector, and Cartesian equations of a plane which passes through the point $(1,4,6)$ and the normal vector to the plane is $(\hat{i}-2 \hat{j}+\hat{k})$. Solution:...
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