Q.Find the vector equation of a plane which is at a distance of 3 units from the origin, $2\hat{i} + 3\hat{j} - 6\hat{k}$ being a normal to the plane. Also get its Cartesian equation.

Showing the 12 most recent of 28 on this concept.

1. CBSE 2026Set ANNUAL1 markMCQ
   Q.The equation of the plane having intercepts 2, −3, 1 on X, Y and Z axis respectively is ................. .

   (a) 2x − 3y + z = 0

   (b) 3x − 2y + 6z = 6

   (c) 3x + 2y + z = 3

   (d) x + 2y + 3z = 2

   ›Reveal solutionSolution

   Use the intercept form of a plane: $x/a+y/b+z/c=1$.

   With $a=2,\,b=-3,\,c=1$:

   $\dfrac{x}{2}+\dfrac{y}{-3}+\dfrac{z}{1}=1$

   …

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2. CBSE 2025Set E1 markMCQ
   Q.Equation of a plane parallel to the plane $9x - 8y + 7z = 10$ is

   (a) $9x - 8y - 7z = 5$

   (b) $9x - 8y + 7z = 5$

   (c) $9x + 8y + 7z = 5$

   (d) $9x - y + 7z = 5$

   ›Reveal solutionSolution

   Parallel planes have identical coefficients of $x,y,z$ (same normal), only the constant differs.

   The plane $9x-8y+7z=10$ has normal direction $(9,-8,7)$. Any parallel plane must have the same normal, i.e. the same coefficients of $x, y, z$, differi …

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3. CBSE 2024Set D1 markMCQ
   Q.The equation of the $xy$-plane is

   (a) $x = 0$

   (b) $y = 0$

   (c) $z = 0$

   (d) none of these

   ›Reveal solutionSolution

   Points on the $xy$-plane have $z=0$.

   The $xy$-plane consists of all points $(x, y, 0)$; the defining condition is that the $z$-coordinate vanishes. Hence its equation is $z = 0$. (Simila …

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4. CBSE 2024Set D1 markMCQ
   Q.The equation of the plane parallel to the plane $3x - 5y + 4z = 11$ is

   (a) $3x - 5y + 4z = 21$

   (b) $3x + 5y + 4z = 25$

   (c) $3x + 5y + 4z = 35$

   (d) none of these

   ›Reveal solutionSolution

   Parallel planes share the same normal, so the coefficients of $x, y, z$ must be identical: $3x - 5y + 4z = 21$.

   Two planes are parallel iff their normal vectors are proportional. The given plane $3x - 5y + 4z = 11$ has normal $(3, -5, 4)$. A parallel plane must therefore have the form

   $3x - 5y + 4z = k$ …

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5. CBSE 2024Set ANNUAL1 markMCQ
   Q.The Cartesian equation of the plane r⃗ · (î + ĵ − k̂) = 2 is :

   (a) x − y − z = 2

   (b) x + y − z = 2

   (c) x + y + z = 2

   (d) x + y − z = −2

   ›Reveal solutionSolution

   Writing r⃗ = xî + yĵ + zk̂ and dotting with (î + ĵ − k̂) directly gives the Cartesian form x + y − z = 2.

   The vector equation of a plane is $\vec r \cdot \vec n = d$, where $\vec r = x\hat\imath + y\hat\jmath + z\hat k$ is the position vector of a general point and $\vec n$ is the plane's normal.

   Here $\vec n = \hat\imath + \hat\jmath - \hat k$ and $d = 2$.

   …

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6. CBSE 2023Set ANNUAL1 markMCQ
   Q.The Cartesian equation of the plane $\vec{r}\cdot(\hat{i}+\hat{j}-\hat{k}) = 2$ is-

   (a) $x+y-z=0$

   (b) $x+y-z=2$

   (c) $x+y-z=1$

   (d) $x+y+z+2=0$

   ›Reveal solutionSolution

   Substitute $\vec r=x\hat i+y\hat j+z\hat k$ into the vector equation of the plane and take the dot product.

   Given $\vec r\cdot(\hat i+\hat j-\hat k)=2$, with $\vec r=x\hat i+y\hat j+z\hat k$:

   …

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7. CBSE 2023Set E1 markMCQ
   Q.Direction ratios of the normal to the plane $x+2y-3z+15=0$ are

   (a) $1,2,3$

   (b) $1,-2,3$

   (c) $1,2,-3$

   (d) $1,2,15$

   ›Reveal solutionSolution

   The normal to $ax+by+cz+d=0$ has direction ratios $a,b,c$, i.e. $1,2,-3$.

   For a plane written as $ax+by+cz+d=0$, the normal vector is $a\vec{i}+b\vec{j}+c\vec{k}$, so its direction ratios are the coe …

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8. CBSE 2023Set E1 markMCQ
   Q.Equation of a plane parallel to the plane $x-8y-9z=12$ is

   (a) $x+8y+9z=12$

   (b) $x-8y-9z=2023$

   (c) $8x-y-9z=12$

   (d) $x-9y-8z=12$

   ›Reveal solutionSolution

   A plane parallel to $x-8y-9z=12$ keeps the coefficients $1,-8,-9$; only the constant changes, e.g. $x-8y-9z=2023$.

   Two planes are parallel iff their normal vectors are proportional, i.e. the coefficients of $x,y,z$ are the same (up to a common factor). Only the constant term may differ.

   …

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9. CBSE 2023Set ANNUAL1 markMCQ
   Q.The equation of the plane with intercepts of 2, 3 and 4 on the $x, y$ and $z$-axes respectively is:

   (a) $4x+6y+3z=12$

   (b) $6x+4y+3z=12$

   (c) $3x+4y+6z=12$

   (d) $5x+4y+3z=0$

   ›Reveal solutionSolution

   Use the intercept form of a plane, $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$, then clear denominators.

   With intercepts $a=2,\ b=3,\ c=4$:

   $\dfrac{x}{2}+\dfrac{y}{3}+\dfrac{z}{4}=1$

   …

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10. CBSE 2023Set ANNUAL1 mark
    Q.Find the intercepts cut off by the plane $2x+y-z=5$ on co-ordinate axes.

    ›Reveal solutionSolution

    Rewrite the plane equation in intercept form $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ by dividing through so the RHS becomes 1.

    $2x+y-z=5$

    Divide both sides by $5$:

    $\dfrac{x}{5/2}+\dfrac{y}{5}+\dfrac{z}{-5}=1$

    …

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11. CBSE 2022Set HE2191 mark
    Q.Write true or false: Equation of a plane in normal form is $lx + my + nz = d$.

    ›Reveal solutionSolution

    This is exactly the standard normal (or perpendicular) form of the equation of a plane.

    The equation of a plane in normal form is $lx+my+nz=d$, where $(l,m,n)$ are the direction cosines of the normal to the plane from the origin, and $d\ (\ge 0)$ is th …

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12. CBSE 2022Set HE2191 mark
    Q.Write true or false: The planes $2x - y + 4z = 5$ and $5x - 2.5y + 10z = 6$ are parallel.

    ›Reveal solutionSolution

    Two planes are parallel if their normal vectors are proportional; check the ratio of coefficients.

    Plane 1: $2x-y+4z=5$, normal $(2,-1,4)$.

    Plane 2: $5x-2.5y+10z=6$, normal $(5,-2.5,10)$.

    …

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