Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
The equations of the given lines can be re-written as\[\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5}\] and \[\frac{x - 8}{7} = \frac{y - 4}{1} = \frac{z - 5}{3}\] We know that the lines \[\frac{x - x_1}{l_1} = \frac{y - y_1}{m_1} = \frac{z - z_1}{n_1}\] and
\[ l_1 = 4, m_1 = 4, n_1 = - 5, l_2 = 7, m_2 = 1, n_2 = 3\]
\[ = \begin{vmatrix}8 - 5 & 4 - 7 & 5 - \left( - 3 \right) \\ 4 & 4 & - 5 \\ 7 & 1 & 3\end{vmatrix}\]
\[ = \begin{vmatrix}3 & - 3 & 8 \\ 4 & 4 & - 5 \\ 7 & 1 & 3\end{vmatrix}\]
\[ = 3\left( 12 + 5 \right) + 3\left( 12 + 35 \right) + 8\left( 4 - 28 \right)\]
\[ = 51 + 141 - 192\]
\[ = 0\]
So, the given lines are coplanar.
APPEARS IN
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 points.
(−5, 0, −6), (−3, 10, −9) and (−2, 6, −6)
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)
Show that the four points (0, −1, −1), (4, 5, 1), (3, 9, 4) and (−4, 4, 4) are coplanar and find the equation of the common plane.
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 equations of the following planes in scalar product form \[\left( \vec{r} \cdot \vec{n} = d \right):\]\[\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( 4 \hat{i} - 2 \hat{j} + 3 \hat{k} \right)\]
Find the Cartesian forms of the equations of the following planes. \[\vec{r} = \left( \hat{i} - \hat{j} \right) + s\left( - \hat{i} + \hat{j} + 2 \hat{k} \right) + t\left( \hat{i} + 2 \hat{j} + \hat{k} \right)\]
Find the Cartesian forms of the equations of the following planes.
Find the vector equation of the following planes in non-parametric form. \[\vec{r} = \left( 2 \hat{i} + 2 \hat{j} - \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \mu\left( 5 \hat{i} - 2 \hat{j} + 7 \hat{k} \right)\]
Find the equation of the plane which is parallel to 2x − 3y + z = 0 and which passes through (1, −1, 2).
Find the equation of the plane through (3, 4, −1) which is parallel to the plane \[\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + 2 = 0 .\]
Find the equation of the plane passing through the line of intersection of the planes 2x − 7y + 4z − 3 = 0, 3x − 5y + 4z + 11 = 0 and the point (−2, 1, 3).
Show that the lines \[\vec{r} = \left( 2 \hat{j} - 3 \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)\] are coplanar. Also, find the equation of the plane containing them.
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 \[\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}\] 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.
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.
The equation of the circle passing through the foci of the ellipse `x^2/16 + y^2/9` = 1 and having centre at (0, 3) is
Find the equations of the planes that passes through three points (1, 1, – 1), (6, 4, – 5),(– 4, – 2, 3)
