Advertisements
Advertisements
प्रश्न
Find the vector equations of the coordinate planes.
उत्तर
\[ \text{ Vector equation of XY-plane}\]
\[\text{ This plane is passing through the origin whose position vector is } \vec{a} = 0^\to \text{ and perpendicular to z-axis whose position vector is } \hat{k} .\]
\[\text{ So, the equation of the XY-plane is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \hat{k} = \vec{0} . \hat{k} \]
\[ \Rightarrow \vec{r} . \hat{k} = 0\]
\[\text{ Vector equation of YZ-plane } \]
\[\text{ This plane is passing through the origin whose position vector is } a^\to = 0^\to \text{ and perpendicular tox-axis whose position vector is } \hat{i} .\]
\[\text{ So, the equation of the YZ-plane is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \hat{i} = \vec{0} . \hat{i} \]
\[ \Rightarrow \vec{r} . \hat{i} = 0\]
\[\text{ Vector equation of XZ-plane } \]
\[ \text{ This plane is passing through the origin whose position vector is } \vec{a} = \vec{0} \text{ and perpendicular toy-axis whose position vector is } \hat{j} .\]
\[\text{ So, the equation of the XZ-plane is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \hat{j} = \vec{0} . \hat{j} \]
\[ \Rightarrow \vec{r} . \hat{j} = 0\]
APPEARS IN
संबंधित प्रश्न
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.
If the points (1, 1, p) and (−3, 0, 1) be equidistant from the plane `vecr.(3hati + 4hatj - 12hatk)+ 13 = 0`, then find the value of p.
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.
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes `vecr = (hati - hatj + 2hatk) = 5`and `vecr.(3hati + hatj + hatk) = 6`
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
\[\vec{r} \cdot \left( - \hat{i} + \hat{j} + 2 \hat{k} \right) = 9\]
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
Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.
Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).
Find the vector equation of the plane passing through points A (a, 0, 0), B (0, b, 0) and C(0, 0, c). Reduce it to normal form. If plane ABC is at a distance p from the origin, prove that \[\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} .\]
Determine the value of λ for which the following planes are perpendicular to each other.
Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y − z = 1 and 3x − 4y + z = 5.
Find the equation of the plane passing through (a, b, c) and parallel to the plane \[\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .\]
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 coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x + y + z = 7.
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 image of the point (0, 0, 0) in the plane 3x + 4y − 6z + 1 = 0.
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 from the point (1, 1, 2) to the plane 2x − 2y + 4z + 5 = 0. Also, find the length of the perpendicular.
Find the image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0.
Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane \[\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 4 \hat{k} \right) + 5 = 0 .\]
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 the YOZ- plane and passing through (−4, 1, 0).
Write the equation of the plane passing through points (a, 0, 0), (0, b, 0) and (0, 0, c).
Write the general equation of a plane parallel to X-axis.
Write the value of k for which the planes x − 2y + kz = 4 and 2x + 5y − z = 9 are perpendicular.
Write the equation of the plane \[\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}\] in scalar product form.
Write the equation of the plane containing the lines \[\vec{r} = \vec{a} + \lambda \vec{b} \text{ and } \vec{r} = \vec{a} + \mu \vec{c} .\]
Find the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane \[\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2\]
The vector equation of the plane containing the line \[\vec{r} = \left( - 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + \lambda\left( 3 \hat{i} - 2 \hat{j} - \hat{k} \right)\] and the point \[\hat{i} + 2 \hat{j} + 3 \hat{k} \] is
The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is
Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.
Find the vector equation of the plane which contains the line of intersection of the planes `vec("r").(hat"i"+2hat"j"+3hat"k"),-4=0, vec("r").(2hat"i"+hat"j"-hat"k")+5=0`and which is perpendicular to the plane`vec("r").(5hat"i"+3hat"j"-6hat"k"),+8=0`
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.
Show that the points `(hat"i" - hat"j" + 3hat"k")` and `3(hat"i" + hat"j" + hat"k")` are equidistant from the plane `vec"r" * (5hat"i" + 2hat"j" - 7hat"k") + 9` = 0 and lies on opposite side of it.
The point at which the normal to the curve y = `"x" + 1/"x", "x" > 0` is perpendicular to the line 3x – 4y – 7 = 0 is:
