Find the vector and Cartesian equations of a plane
Question: Find the vector and Cartesian equations of a plane which is at a distance of 6 units from the origin and which has a normal with direction ratios $2,-1,-2$. Solution:...
Read More →Find the vector and Cartesian equations of a plane which is at a distance
Question: Find the vector and Cartesian equations of a plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and whose normal vector from the origin is $(2 \hat{i}-3 \hat{j}+4 \hat{k})$. Solution:...
Read More →Find the vector and Cartesian equations of a plane which is at a distance
Question: Find the vector and Cartesian equations of a plane which is at a distance of 7 units from the origin and whose normal vector from the origin is $(3 \hat{i}+5 \hat{j}-6 \hat{k})$. Solution:...
Read More →Find the vector and Cartesian equations of a plane which is at a distance
Question: Find the vector and Cartesian equations of a plane which is at a distance of 5 units from the origin and which has $\hat{k}$ as the unit vector normal to it. Solution:...
Read More →Solve this following
Question: If $O$ is the origin and $P(1,2,-3)$ be a given point, then find the equation of the plane passing through $P$ and perpendicular to $O P$. 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 $(1,2,3)$ and perpendicular to a line with direction ratios $2,3,-$ $4 .$ Solution:...
Read More →A plane meets the coordinate axes at
Question: A plane meets the coordinate axes at $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ respectively such that the centroid of $\triangle \mathrm{ABC}$ is $(1,-2,3)$. Find the equation of the plane. Solution:...
Read More →Find the equation of the plane which passes through the point
Question: Find the equation of the plane which passes through the point $(2,-3,7)$ and makes equal intercepts on the coordinate axes. Solution:...
Read More →Reduce the equation of the plane
Question: Reduce the equation of the plane $4 x-3 y+2 z=12$ to the intercept form, and hence find the intercepts made by the plane with the coordinate axes. Solution: $\therefore \frac{x}{3}+\frac{y}{-4}+\frac{z}{6}=1$...
Read More →Write the equation of the plane whose intercepts on the coordinate axes are
Question: Write the equation of the plane whose intercepts on the coordinate axes are $2,-4$ and 5 respectively. Solution:...
Read More →Show that the four points
Question: Show that the four points $A(0,-1,0), B(2,1,-1), C(1,1,1)$ and $D(3,3,0)$ are coplanar. Find the equation of the plane containing them. Solution:...
Read More →Show that the four points
Question: Show that the four points $A(3,2,-5), B(-1,4,-3), C(-3,8,-5)$ and $D(-3,2,1)$ are coplanar. Find the equation of the plane containing them. Solution:...
Read More →Find the equation of the plane passing through each group of points:
Question: Find the equation of the plane passing through each group of points: (i) $\mathrm{A}(2,2,-1), \mathrm{B}(3,4,2)$ and $\mathrm{C}(7,0,6)$ (ii) $\mathrm{A}(0,-1,-1), \mathrm{B}(4,5,1)$ and $\mathrm{C}(3,9,4)$ (iii) $\mathrm{A}(-2,6,-6), \mathrm{B}(-3,10,9)$ and Solution:...
Read More →Solve this following
Question: If the points $A(-1,3,2), B(-4,2,-2)$ and $C(5,5, \lambda)$ are collinear then the value of $\lambda$ is A. 5 B. 7 C. 8 D. 10 Solution:...
Read More →Solve this following
Question: If $\left(a_{1}, b_{1}, c_{1}\right)$ and $\left(a_{2}, b_{2}, c_{2}\right)$ be the direction ratios of two parallel lines then A. $\mathrm{a}_{1}=\mathrm{a}_{2}, \mathrm{~b}_{1}=\mathrm{b}_{2}, \mathrm{c}_{1}=\mathrm{c}_{2}$ B. $\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{~b}_{2}}=\frac{\mathrm{c}_{1}}{\mathrm{c}_{2}}$ C. $a_{1}^{2}+b_{1}^{2}+c_{1}^{2}=a_{2}^{2}+b_{2}^{2}+c_{2}^{2}$ D. $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$ Solution:...
Read More →Solve this following
Question: If a line makes angles $a, \beta$ and $y$ with the $x$-axis, $y$-axis and $z$-axis respectively then $\left(\sin ^{2} a+\sin ^{2} \beta\right.$ $\left.+\sin ^{2} y\right)=?$ A. 1 B. 3 C. 2 D. $\frac{3}{2}$ Solution:...
Read More →The straight line
Question: The straight line $\frac{x-2}{3}=\frac{y-3}{1}=\frac{z+1}{0}$ is A. parallel to the $x$-axis B. parallel to the $y$-axis C. parallel to the z-axis D. perpendicular to the z-axis Solution:...
Read More →The angle between two lines having direction ratios
Question: The angle between two lines having direction ratios $1,1,2$ and $(\sqrt{3}-1),(-\sqrt{3}-1), 4$ is A. $\frac{\pi}{6}$ B. [] $\frac{\pi}{2}$ C. $\frac{\pi}{3}$ D. $\frac{\pi}{4}$ Solution:...
Read More →The Cartesian equations of a lines are
Question: The Cartesian equations of a lines are $\frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}+1}{3}=\frac{\mathrm{z}-3}{-2}$. What is its vector equation? A. $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ B. $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ C. $\overrightarrow{\mathrm{r}}=(2 \h...
Read More →The vector equation of the x-axis is given by
Question: The vector equation of the $x$-axis is given by A. $\overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}$ B. $\vec{r}=\hat{j}+\hat{k}$ C. $\overrightarrow{\mathrm{r}}=\lambda \hat{\mathrm{i}}$ D. none of these Solution:...
Read More →The coordinates of the point where the line through the points
Question: The coordinates of the point where the line through the points $A(5,1,6)$ and $B(3,4,1)$ crosses the yz-plane is A. $(0,17,-13)$ B. $\left(0, \frac{-17}{2}, \frac{13}{2}\right)$ C. $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$ D. none of these Solution:...
Read More →A line passes through the point
Question: A line passes through the point $A(-2,4,-5)$ and is parallel to the line $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$. The vector equation of the line is A. $\overrightarrow{\mathrm{r}}=(-3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-8 \hat{\mathrm{k}})+\lambda(-2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ B. $r=(-2 \hat{\imath}+4 \hat{\jmath}-5 \hat{k})+\lambda(3 \hat{\imath}+5 \hat{\jmath}+6 \hat{k})$ C.$\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+6 \hat{\...
Read More →The Cartesian equations of a line are
Question: The Cartesian equations of a line are $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+2}{3}=\frac{\mathrm{z}-5}{-1}$. Its vector equation is A. $\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ B. $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$ C. $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i...
Read More →A line passes through the point
Question: A line passes through the point $\mathrm{A}(5,-2,4)$ and it is parallel to the vector $(2 \hat{\mathrm{j}}-\hat{\mathrm{j}}+3 \hat{\mathrm{K}})$. The vector equation of the line is A. $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$ B. $\overrightarrow{\mathrm{r}}=(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}...
Read More →A line is perpendicular to two lines having direction ratios
Question: A line is perpendicular to two lines having direction ratios $1,-2,-2$ and $0,2,1$. The direction cosines of the line are A. $\frac{-2}{3}, \frac{1}{3}, \frac{2}{3}$ B. $\frac{2}{3}, \frac{1}{3}, \frac{-1}{3}$ C. $\frac{2}{3}, \frac{-1}{3}, \frac{2}{3}$ D. none of these Solution:...
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