Recent questions and answers in Vector Algebra

Show that the points $(1 ,3 ,1), (1, 1, -1),(-1,1, 1),(2 ,2,- 1)$ are lying on the same plane.(Hint : It is enough to prove any three vectors formed by these four points are coplanar).

answered Feb 15, 2020 by sachinbista868

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The constant forces $\overrightarrow{2i}-\overrightarrow{5j}+ \overrightarrow{6k}, -\overrightarrow{i}+\overrightarrow{2j}-\overrightarrow{k},$ and $\overrightarrow{2i}+\overrightarrow{7j}$ act on a particle which is displaced from position $\overrightarrow{4i}-\overrightarrow{3j}-\overrightarrow{2k}$ to position $\overrightarrow{6i}+\overrightarrow{j}-\overrightarrow{3k}$ . find the work done.

answered Jun 20, 2013 by thanvigandhi_1

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Show that diameter of a sphere subtends a right angle at a point on the surface.

answered Jun 18, 2013 by thanvigandhi_1

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Find the centre and radius of the following spheres: $\overrightarrow{r}^{2}-\overrightarrow{r} . (\overrightarrow{4i}+\overrightarrow{2j}-\overrightarrow{6k})-11=0$

answered Jun 18, 2013 by thanvigandhi_1

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Find the centre and radius of the following spheres : $x^{2}+y^{2}+z^{2}+4x-8x+2z=5$

answered Jun 18, 2013 by thanvigandhi_1

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Find the centre and radius of the following spheres : $|\overrightarrow{2r}+(\overrightarrow{3i}-\overrightarrow{j}+\overrightarrow{4k})|=4$

answered Jun 18, 2013 by thanvigandhi_1

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Find the centre and radius of the following spheres : $| \overrightarrow{r}-(\overrightarrow(2i)-\overrightarrow(j)+\overrightarrow{4k})|=5$

answered Jun 18, 2013 by thanvigandhi_1

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If $A (-1 , 4 , -3 )$ is one end of a diameter $AB$ of the sphere $x^{2}+y^{2}+z^{2}-3x-2y+2z-15=0$ than, Find the coordinates of $B$ .

answered Jun 18, 2013 by thanvigandhi_1

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Obtain the vector and cartesian equation of thesphere whose centre is $(1 , -1 , 1 )$ and radius is the same as that of the sphere $|\overrightarrow{r}-(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{2k})|=5.$

answered Jun 18, 2013 by thanvigandhi_1

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Find the vector and cartesian equationof the sphere on the join of the points $A$ and $B$ having position vectors $\overrightarrow{2i}+\overrightarrow{6j}-\overrightarrow{7k}$ and $-\overrightarrow{2i}+\overrightarrow{4j}-\overrightarrow{3k}$ respectively as a diameter . Find also the centre and radius of the sphere.

answered Jun 18, 2013 by thanvigandhi_1

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Find the vector equation of a sphere with centre having position vector $\overrightarrow{2i}-\overrightarrow{j}+\overrightarrow{3k}$ and radius $4$ units. Also find the equation in cartesian form.

answered Jun 18, 2013 by thanvigandhi_1

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Find the angle between the line $\overrightarrow{r}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{3k}+\lambda(\overrightarrow{2i}+\overrightarrow{j}-\overrightarrow{k})$ and the plane $\overrightarrow{r}.(\overrightarrow{i}+\overrightarrow{j})=1.$

answered Jun 18, 2013 by thanvigandhi_1

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Find the angle between the line $\large\frac{x-2}{3}=\frac{y+1}{-1}=\frac{z-3}{-2}$ and the plane $3x+4y+z+5=0$

answered Jun 18, 2013 by thanvigandhi_1

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The planes $\overrightarrow{r} .(\overrightarrow{2i}+\lambda\overrightarrow{j}-\overrightarrow{3k})=10$ and $\overrightarrow{r}. (\lambda\overrightarrow{i}+\overrightarrow{3j}+\overrightarrow{k})=5$ are perpendiculare find $\lambda$

answered Jun 17, 2013 by thanvigandhi_1

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Show that the following planes are at right angles. $\overrightarrow{r}(\overrightarrow{2i}-\overrightarrow{j}+\overrightarrow{k})=15$ and $\overrightarrow{r}(\overrightarrow{i}-\overrightarrow{j}-\overrightarrow{3k})=3$

answered Jun 17, 2013 by thanvigandhi_1

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Find the angle between following plane: $r=(3i+j-k)=7$ and $r=(i+4j-2k)=10$

answered Jun 17, 2013 by thanvigandhi_1

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Find the angle between following planes; $2x-3y+4z=1$ and $-x+y=4$

answered Jun 17, 2013 by thanvigandhi_1

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Find the angle between the followin planes: $2x+y-z=9$ and $x+2y+z=7$

answered Jun 17, 2013 by thanvigandhi_1

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Find the distane between the parallel planes $x-y+3z+5=0; 2x-2y+6z+7=0$

answered Jun 17, 2013 by thanvigandhi_1

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Find the distance from the origin to the plane $\overrightarrow{r}=(\overrightarrow{2i}+\overrightarrow{j}+\overrightarrow{5j})=7$

answered Jun 17, 2013 by thanvigandhi_1

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Find the meeting point of the line $\overrightarrow{r}(\overrightarrow{2i}+\overrightarrow{j}-\overrightarrow{3k}) + t(\overrightarrow{2i}-\overrightarrow{j}-\overrightarrow{k})$ and the plane. $x-2y+3z+7=0$

answered Jun 17, 2013 by thanvigandhi_1

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Find the point of intersection of the line $\overrightarrow{r}=(\overrightarrow{j}-\overrightarrow{k})+s(\overrightarrow{2i}-\overrightarrow{j}+\overrightarrow{k})$ and $x z$ - plane.

answered Jun 17, 2013 by thanvigandhi_1

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Can u draw a plane through the given two lines? justify your answer. $\overrightarrow{r}(\overrightarrow{i}+\overrightarrow{2j}-\overrightarrow{4k})+t (\overrightarrow{2i}+\overrightarrow{3j} +\overrightarrow{6k})$ and $r=(\overrightarrow{3i}+\overrightarrow{3j}-\overrightarrow{5k})+ s(-\overrightarrow{2i}+\overrightarrow{3j}+\overrightarrow{8k})$

answered Jun 17, 2013 by thanvigandhi_1

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Find the equations of the plane which contains the two lines $\large\frac{x+1}{2}=\frac{y-2}{-3}=\frac{z-3}{4}$ and $\large\frac{x-4}{3}=\frac{y-1}{2}=z-8$

answered Jun 16, 2013 by thanvigandhi_1

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Find the cartesian form of following planes: $\overrightarrow{r}=(1+s+t) \overrightarrow{i}+(2-s+t)\overrightarrow{j}+(3-2s+2t)\overrightarrow{k}$

answered Jun 16, 2013 by thanvigandhi_1

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Find the cartesian form of following planes: $\overrightarrow{r}=(s-2t) \overrightarrow{i}+(3-t)\overrightarrow{j}+(2s+t)\overrightarrow{k}$

answered Jun 16, 2013 by thanvigandhi_1

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Derive the equation of the plane in the intercept form.

answered Jun 16, 2013 by thanvigandhi_1

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Find the vector cartesian equation of the plane passing through the points with position vectors $\overrightarrow{3i}+\overrightarrow{4j}+\overrightarrow{2k}, \overrightarrow{2i}-\overrightarrow{2j}-\overrightarrow{2k},$ and $\overrightarrow{7i}+\overrightarrow{k}.$

answered Jun 16, 2013 by thanvigandhi_1

1 answer

Find the vector cartesian equation of the plane containing the line $\large\frac{x-2}{2}=\frac{y-2}{3}=\frac{z-1}{-2}$ and passing through the point $(-1 , 1 , -1 )$

answered Jun 16, 2013 by thanvigandhi_1

1 answer

Find the vector and cartesian equations of the plane through the points $(1 , 2 , 3 )$ and $(2 , 3 , 1 )$ perpendicular to the plane $3x-2y+4z-5=0$

answered Jun 16, 2013 by thanvigandhi_1

1 answer

Find the vector and cartesian equations of the plane passing through the points $A(1 , -2 , 3 )$ and $B(-1 , 2 , -1 )$ and is parallel to the line $\large\frac{x-2}{2}=\frac{y+1}{3}=\frac{z-1}{4}.$

answered Jun 16, 2013 by thanvigandhi_1

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Find the vector and cartesian equation to the plane through the point $(-1 , 3 , 2 )$ and perpendicular to the planes $x+2y+2z=5$ and $3x+y+2z=8.$

answered Jun 14, 2013 by thanvigandhi_1

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Find the vector and cartesian equations of the plane through the point $(1 , 3 , 2 )$ and parallel to the lines $\large\frac{x+1}{2}=\frac{y+2}{-1}=\frac{z+3}{3}$ and parallel to the line $\large\frac{x-2}{1}=\frac{y+1}{2}=\frac{z+2}{2}$

answered Jun 14, 2013 by thanvigandhi_1

1 answer

Find the vector and cartesian equations of the plane containing the line $\Large\frac{x-2}{2}=\frac{y-2}{3}=\frac{z-1}{3}$ and parallel to the line $\Large\frac{x+1}{3}=\frac{y-1}{2}=\frac{z+1}{1}$

answered Jun 14, 2013 by thanvigandhi_1

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Find the vector and cartesian equations of the plane through the point $(2 , -1 , 4 )$ and parallel to the plane $\overrightarrow{r} . (\overrightarrow{4i}-\overrightarrow{12j}-\overrightarrow{3k})=7.$

answered Jun 14, 2013 by thanvigandhi_1

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Find the equation of the plane through the point whose $p.v.$ is $\overrightarrow{2i}-\overrightarrow{j}+\overrightarrow{k}$ and perpendicular to the vector $\overrightarrow{4i}+\overrightarrow{2j}-\overrightarrow{3k}$

answered Jun 14, 2013 by thanvigandhi_1

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The foot of the perpendicular drawn from the origin to the plane is $(8 , -4 , 3 )$ find the equation of the plane.

answered Jun 14, 2013 by thanvigandhi_1

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Find the length of the perpendicular frome the origin to the plane $\overrightarrow{r} . (\overrightarrow{3i}+\overrightarrow{4j}+\overrightarrow{12k})=26$ .

answered Jun 14, 2013 by thanvigandhi_1

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Find the unit normal vectors to the plane $2x-y+2z=5$

answered Jun 14, 2013 by thanvigandhi_1

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Find the vector and cartesian equations of a plane which is at a distance of $18$ units from the origin and which is normal to the vector $\overrightarrow{2i}+\overrightarrow{7j}+\overrightarrow{8k}$

answered Jun 14, 2013 by thanvigandhi_1

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If the point $(\lambda , 0 , 3 ), (1 , 3 , -1 )$ and $(-5 , -3 , 7 )$ are collinear than find $\lambda$ .

answered Jun 14, 2013 by thanvigandhi_1

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Show that $(2 , -1 ,3 ),(1 ,-1, 0 )$ and $(3, -1, 6 )$ are collinear.

answered Jun 12, 2013 by thanvigandhi_1

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Find the shortest distance between the skew lines $\large\frac{x-6}{3}=\frac{y-7}{-1}=\frac{z-4}{1}$ and $\large\frac{x}{-3}=\frac{y+9}{2}=\frac{z-2}{4}$

answered Jun 12, 2013 by thanvigandhi_1

1 answer

Show that the lines $\large \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z}{3}$ and $\large\frac{x-2}{1}=\frac{y-1}{2}=\frac{-z-1}{1}$ intersect and find their point of intersection.

answered Jun 12, 2013 by thanvigandhi_1

1 answer

Show that the following two lines are skew lines: $\overrightarrow{r}=(\overrightarrow{3i}+\overrightarrow{5j}+\overrightarrow{7k})+ t (\overrightarrow{i}-\overrightarrow{2j}+\overrightarrow{k})$ and $\overrightarrow{r}=(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}) + s (\overrightarrow{7i}-\overrightarrow{6j}+\overrightarrow{7k})$

answered Jun 12, 2013 by thanvigandhi_1

1 answer

Find the shortest distance between the parallel lines $\large\frac{x-1}{-1}=\frac{y}{3}=\frac{z+3}{2}$ and $\large\frac{x-3}{-1}=\frac{y+1}{3}=\frac{z-1}{2}$

answered Jun 12, 2013 by thanvigandhi_1

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Find the shortest distance between the parallel lines $\overrightarrow{r}=(\overrightarrow{2i}-\overrightarrow{j}-\overrightarrow{k}) + t (\overrightarrow{i}-\overrightarrow{2j}+\overrightarrow{3k})$ and $\overrightarrow{r}=(\overrightarrow{i}+\overrightarrow{2j}+\overrightarrow{k}) + s (\overrightarrow{i}-\overrightarrow{2j}+\overrightarrow{3k})$

answered Jun 12, 2013 by thanvigandhi_1

1 answer

Find the angle between the lines $\overrightarrow{r}=\overrightarrow{5i}-\overrightarrow{7j}+\mu (-\overrightarrow{i}+\overrightarrow{4j}+\overrightarrow{2k}) \overrightarrow{r}=-\overrightarrow{2i}+\overrightarrow{k}+\lambda (\overrightarrow{3i}+\overrightarrow{4k})$

answered Jun 12, 2013 by thanvigandhi_1

1 answer

Find the angle between the following lines. $\large\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-4}{6}$ and $x+1=\large\frac{y+2}{2}=\frac{z-4}{2}$

answered Jun 12, 2013 by thanvigandhi_1

1 answer

Find the vector and cartesian equation of the line joining the points $(1 , -2 , 1 )$ and $(0 , -2 , 3 ).$

answered Jun 12, 2013 by thanvigandhi_1

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