Solution: (a) The distance of the point $(0,0,0)$ from the plane $3 x-4 y+12=3 \Rightarrow$ $3 x-4 y+12 z-3=0$ is $\begin{array}{l} \frac{\left|a x_{1}+b y_{1}+c...
In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...
Find the angle between the planes whose vector equations are $\vec{r}:(2 \hat{i}+2 \hat{j}-3 \hat{k})=5, \vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$
Solution: It is given that The eq. of the given planes are $\vec{r}(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 \text { and } \vec{r}(3 \hat{i}-3 \hat{j}+5 \hat{k})=5$ If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$...
Find the vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=7, \vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})=9$ and through the point $(2,1,3)$
Solution: Let's consider the vector eq. of the plane passing through the intersection of the planes are $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7...
Find the intercepts cut off by the plane 2x + y – z = 5.
Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...
Find the vector and Cartesian equations of the planes
(a) that passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{i}+\hat{j}-\hat{k}$
(b) that passes through the point $(1,4,6)$ and the normal vector to the plane is $\hat{i}-2 \hat{j}+\hat{k}$
Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) x + y + z = 1
(b) 5y + 8 = 0
Solution: (a) $x+y+z=1$ Let the coordinate of the foot of $\perp \mathrm{P}$ from the origin to the given plane be $P(x, y, z)$ $x+y+z=1$ The direction ratio are $(1,1,1)$ $\begin{array}{l}...
Find the Cartesian equation of the following planes:
(a) $\overrightarrow{\mathrm{r}} \cdot[(\mathrm{s}-2 \mathrm{t}) \hat{\mathrm{i}}+(3-\mathrm{t}) \hat{\mathrm{j}}+(2 \mathrm{~s}+\mathrm{t}) \hat{\mathrm{k}}]=15$
Solution: Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\overrightarrow{\mathrm{r}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y}...
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$
Solution: It is given that, The vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ Vector equation of the plane with position vector $\overrightarrow{\mathrm{r}}$ is $\vec{r} \cdot...
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z = 2
(b) x + y + z = 1
Solution: (a) $z=2$ It is given that The eq. of the plane, $z=2$ or $0 x+0 y+z=2 \ldots (1) .$ The direction ratio of the normal $(0,0,1)$ Using the formula, $\begin{array}{l}...
In Fig. 11.7, AB is a diameter of the circle, \[AC=6\]cm and \[BC=8\] cm. Find the area of the shaded region (Use \[\pi =3.14\]).
Solution From the given question, \[AC=6\]cm and \[BC=8\] cm We know that a triangle in a semi-circle with hypotenuse as diameter is right angled triangle. By using Pythagoras theorem in triangle...
Find the area of the flower bed (with semi-circular ends) shown in Fig. 11.6.
Solution: From the given figure, We got that the Length and breadth of the rectangular portion (AFDC) of the flower bed are \[38\] cm and \[10\] cm respectively. We know that, Area of the flower bed...
A cow is tied with a rope of length \[14\] m at the corner of a rectangular field of dimensions \[20m\times 16m\]. Find the area of the field in which the cow can graze.
Let us consider ABCD be a rectangular field. Given, Length of the field = \[20\] m Given, Breadth of the field = \[16\] m From the given question, A cow is tied with a rope at a point A. Let us...
The wheel of a motor cycle is of radius \[35\] cm. How many revolutions per minute must the wheel make so as to keep a speed of \[66\] km/h?
From the question Radius of wheel = r = \[35\] cm We know that one revolution of the wheel is equal to Circumference of the wheel i.e., \[2\pi r\] = \[2\times (22/7)\times 35\] = \[220\] cm But,...
Find the area of a sector of a circle of radius \[28\]cm and central angle \[{{45}^{\circ }}\].
We know that Area of a sector of a circle = \[(1/2){{r}^{2}}\theta \], Here r is the radius and \[\theta \] is the angle in radians subtended by the arc at the center of the circle From the given...
In Fig. 11.5, a square of diagonal \[8\] cm is inscribed in a circle. Find the area of the shaded region.
Let us take a be the side of square. From the question we got, diagonal of square and diameter of circle is \[8\] cm In right angled triangle ABC, By Using Pythagoras theorem we got,...
Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii \[15\] cm and \[18\] cm.
Given Radius of first circle = \[{{r}_{1}}\] = \[15\] cm Given Radius of second circle = \[{{r}_{2}}\] = \[18\] cm Therefore, Circumference of first circle of radius \[{{r}_{1}}\]= \[2\pi...