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The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear. - Mathematics

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प्रश्न

The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear.

विकल्प

  • True

  • False

MCQ
सत्य या असत्य

उत्तर

This statement is True.

Explanation:

Let A, B, C be the points (1, 2, 3), (–2, 3, 4) and (7, 0, 1), respectively

Then, the direction ratios of each of the lines AB and BC are proportional to – 3, 1, 1.

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अध्याय 11: Three Dimensional Geometry - Solved Examples [पृष्ठ २३४]

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एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12
अध्याय 11 Three Dimensional Geometry
Solved Examples | Q 23 | पृष्ठ २३४

संबंधित प्रश्न

Find the equations of the planes parallel to the plane x-2y + 2z-4 = 0, which are at a unit distance from the point (1,2, 3).


Find the equation of the plane passing through the following points.

 (2, 1, 0), (3, −2, −2) and (3, 1, 7)


Find the equation of the plane passing through the following point

 (1, 1, 1), (1, −1, 2) and (−2, −2, 2)


Find the equation of the plane passing through the following points. 

(2, 3, 4), (−3, 5, 1) and (4, −1, 2) 

 


Find the equation of the plane passing through the following point

(0, −1, 0), (3, 3, 0) and (1, 1, 1)

 

 


Show that the following points are coplanar.
 (0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0) 


Show that the following points are coplanar. 

 (0, 4, 3), (−1, −5, −3), (−2, −2, 1) and (1, 1, −1)

 

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\] \[\vec{r} = \left( 2 \hat{i} - \hat{k} \right) + \lambda \hat{i} + \mu\left( \hat{i} - 2 \hat{j} - \hat{k}
\right)\]

 

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\] \[\vec{r} = \left( 1 + s - t \right) \hat{t}  + \left( 2 - s \right) \hat{j}  + \left( 3 - 2s + 2t \right) \hat{k} \]

 

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\]\[\vec{r} = \left( \hat{i}  + \hat{j}  \right) + \lambda\left( \hat{i}  + 2 \hat{j}  - \hat{k}  \right) + \mu\left( - \hat{i}  + \hat{j} - 2 \hat{k} \right)\]


Find the vector equation of the following planes in non-parametric form. \[\vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j}  + \left( 2\lambda + \mu \right) \hat{k} \]


Find the vector equation of the following planes in non-parametric form. \[\vec{r} = \left( 2 \hat{i}  + 2 \hat{j}  - \hat{k}  \right) + \lambda\left( \hat{i}  + 2 \hat{j}  + 3 \hat{k}  \right) + \mu\left( 5 \hat{i}  - 2 \hat{j} + 7 \hat{k}  \right)\]

 


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\[\frac{x + 3}{2} = \frac{y - 3}{7} = \frac{z - 2}{5} .\]
  

Show that the lines \[\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1} \text{ and }\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}\]  are coplanar. Also, find the equation of the plane containing them. 

 

Show that the lines  \[\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5}\] and  \[\frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}\] are coplanar.

 

Show that the lines  \[\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}\] and  \[\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}\]  are coplanar. Hence, find the equation of the plane containing these lines.

 

Find the values of  \[\lambda\] for which the lines

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If the lines  \[x =\]  5 ,  \[\frac{y}{3 - \alpha} = \frac{z}{- 2}\] and   \[x = \alpha\] \[\frac{y}{- 1} = \frac{z}{2 - \alpha}\] are coplanar, find the values of  \[\alpha\].

 


If the straight lines  \[\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}\] and \[\frac{x + 1}{2} = \frac{y + 1}{2} = \frac{z}{k}\] are coplanar, find the equations of the planes containing them.

 


Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).


Find the equations of the planes that passes through three points (1, 1, – 1), (6, 4, – 5),(– 4, – 2, 3)


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