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...
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...
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 $\mathrm{n}_{2}$...
and through the point 
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...
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 $x, y$...
and the normal to the plane is 
and the normal vector to the plane is 
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 $(1,0,-2)$...
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}...
![\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](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0789422b9ad2fd168094a22f1dd6043a_l3.png "Rendered by QuickLaTeX.com")
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}...
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...
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}...
![\[AC=6\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9427641f408963e7fa190634f97a222c_l3.png "Rendered by QuickLaTeX.com")
![\[BC=8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bc8f3386b586d61e532feda848c03100_l3.png "Rendered by QuickLaTeX.com")
![\[\pi =3.14\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d3c9ef587e84b10335f84753e7b43347_l3.png "Rendered by QuickLaTeX.com")
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...
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...
![\[14\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-30e216683ad8ac7a76f0fd9b1da36c1e_l3.png "Rendered by QuickLaTeX.com")
![\[20m\times 16m\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7ad7bb661d485a3104c10d7fcf2a1fb1_l3.png "Rendered by QuickLaTeX.com")
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...
![\[35\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4c246379fe948cf04ed07be9d0c37357_l3.png "Rendered by QuickLaTeX.com")
![\[66\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-28337cbee3c1bfe56a8ff193040b617b_l3.png "Rendered by QuickLaTeX.com")
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,...
![\[28\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-28ff37b4edd300cae2d3efd0f2d02387_l3.png "Rendered by QuickLaTeX.com")
![\[{{45}^{\circ }}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-98711490f39e6a1cc04def418479129c_l3.png "Rendered by QuickLaTeX.com")
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...
![\[8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5ba6e07885b354f6c76c81fbd23c392a_l3.png "Rendered by QuickLaTeX.com")
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,...
![\[15\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5ccbb897c59f87c44d67dcb315709226_l3.png "Rendered by QuickLaTeX.com")
![\[18\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fe4a46fb7b7dae2fad6a6d2b9500c5c8_l3.png "Rendered by QuickLaTeX.com")
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...