Advertisements
Advertisements
Question
Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the yz - plane .
Advertisements
Solution
\[ \text{ The equation of the line through the points (5, 1, 6) and (3, 4, 1) is } \]
\[\frac{x - 5}{3 - 5} = \frac{y - 1}{4 - 1} = \frac{z - 6}{1 - 6}\]
\[ \Rightarrow \frac{x - 5}{- 2} = \frac{y - 1}{3} = \frac{z - 6}{- 5}\]
\[\text{ The coordinates of any point on this line are of the form} \]
\[\frac{x - 5}{- 2} = \frac{y - 1}{3} = \frac{z - 6}{- 5} = \lambda\]
\[ \Rightarrow x = - 2\lambda + 5; y = 3\lambda + 1; z = - 5\lambda + 6\]
\[\text{ So, the coordinates of the point on the given line } are\left( - 2\lambda + 5, 3\lambda + 1, - 5\lambda + 6 \right).\]
\[\text{ Since this point lies on the YZ- plane,} \]
\[x = 0\]
\[ \Rightarrow - 2\lambda + 5 = 0\]
\[ \Rightarrow \lambda = \frac{5}{2}\]
\[\text{ So, the coordinates of the point are} \]
\[\left( - 2\lambda + 5, 3\lambda + 1, - 5\lambda + 6 \right)\]
\[ = \left( - 2 \left( \frac{5}{2} \right) + 5, 3 \left( \frac{5}{2} \right) + 1, - 5 \left( \frac{5}{2} \right) + 6 \right)\]
\[ = \left( 0, \frac{17}{2}, \frac{13}{2} \right)\]
APPEARS IN
RELATED QUESTIONS
Find the equations of the planes that passes through three points.
(1, 1, −1), (6, 4, −5), (−4, −2, 3)
Find the equations of the planes that passes through three points.
(1, 1, 0), (1, 2, 1), (−2, 2, −1)
Find the vector equation of a plane passing through a point with position vector \[2 \hat{i} - \hat{j} + \hat{k} \] and perpendicular to the vector \[4 \hat{i} + 2 \hat{j} - 3 \hat{k} .\]
Find the Cartesian form of the equation of a plane whose vector equation is
\[\vec{r} \cdot \left( 12 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + 5 = 0\]
Find the Cartesian form of the equation of a plane whose vector equation is
\[\vec{r} \cdot \left( - \hat{i} + \hat{j} + 2 \hat{k} \right) = 9\]
Find the vector equations of the coordinate planes.
Find the vector equation of each one of following planes.
x + y − z = 5
Find the vector equation of each one of following planes.
x + y = 3
Show that the normals to the following pairs of planes are perpendicular to each other.
x − y + z − 2 = 0 and 3x + 2y − z + 4 = 0
Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined to the coordinate axes.
Find the vector equation of a plane which is at a distance of 3 units from the origin and has \[\hat{k}\] as the unit vector normal to it.
Determine the value of λ for which the following planes are perpendicular to each other.
Obtain the equation of the plane passing through the point (1, −3, −2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.
Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.
Find the equation of the plane passing through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x + 2y + 2z = 5.
Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
Find the equation of a plane passing through the points (0, 0, 0) and (3, −1, 2) and parallel to the line \[\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7} .\]
Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - 5 \hat{k} \right) + 9 = 0 .\]
Find the equation of a plane which passes through the point (3, 2, 0) and contains the line \[\frac{x - 3}{1} = \frac{y - 6}{5} = \frac{z - 4}{4}\] .
Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.
Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line \[\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .\]
Hence, or otherwise, deduce the length of the perpendicular.
Find the length and the foot of perpendicular from the point \[\left( 1, \frac{3}{2}, 2 \right)\] to the plane \[2x - 2y + 4z + 5 = 0\] .
Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5.
Write the equation of the plane parallel to XOY- plane and passing through the point (2, −3, 5).
Write the equation of the plane parallel to the YOZ- plane and passing through (−4, 1, 0).
Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).
Write the distance of the plane \[\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12\] from the origin.
Write the equation of the plane passing through (2, −1, 1) and parallel to the plane 3x + 2y −z = 7.
Find the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane \[\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2\]
Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.
Find a vector of magnitude 26 units normal to the plane 12x − 3y + 4z = 1.
If O be the origin and the coordinates of P be (1, 2,−3), then find the equation of the plane passing through P and perpendicular to OP.
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vec"r".(5hat"i" - 3hat"j" - 2hat"k")` = 38.
