The vector equation of the plane which is at | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The vector equation of a plane at a distance $ d $ from the origin with a unit normal vector $ \hat{n} $ is given by $ \vec{r} \cdot \hat{n} = d $

Vedclass · Accepted Answer

$ \vec{r} \cdot(2 \hat{i}-3 \hat{j}+\hat{k})=3 $

Vedclass · Answer

(A) The vector equation of a plane at a distance $ d $ from the origin with a unit normal vector $ \hat{n} $ is given by $ \vec{r} \cdot \hat{n} = d $. Given the normal vector $ \vec{N} = 2 \hat{i} - 3 \hat{j} + \hat{k} $. First,find the magnitude of $ \vec{N} $: $ |\vec{N}| = \sqrt{2^2 + (-3)^2 + 1^2} = \sqrt{4 + 9 + 1} = \sqrt{14} $. The unit normal vector is $ \hat{n} = \frac{\vec{N}}{|\vec{N}|} = \frac{2 \hat{i} - 3 \hat{j} + \hat{k}}{\sqrt{14}} $. The distance from the origin is $ d = \frac{3}{\sqrt{14}} $. Substituting these into the equation $ \vec{r} \cdot \hat{n} = d $: $ \vec{r} \cdot \left( \frac{2 \hat{i} - 3 \hat{j} + \hat{k}}{\sqrt{14}} \right) = \frac{3}{\sqrt{14}} $. Multiplying both sides by $ \sqrt{14} $,we get: $ \vec{r} \cdot (2 \hat{i} - 3 \hat{j} + \hat{k}) = 3 $.

The vector equation of the plane which is at a distance of $ \frac{3}{\sqrt{14}} $ from the origin and the normal vector from the origin is $ 2 \hat{i}-3 \hat{j}+\hat{k} $ is:

Similar Questions

Let $P$ be the plane passing through the points $(5,3,0), (13,3,-2)$ and $(1,6,2)$. For $\alpha \in N$,if the distances of the points $A(3,4,\alpha)$ and $B(2,\alpha,a)$ from the plane $P$ are $2$ and $3$ respectively,then the positive value of $a$ is:

If $A(-1, 2, 3)$,$B(1, 1, 1)$ and $C(2, -1, 3)$ are points on a plane,then a unit normal vector to the plane $ABC$ is:

$A$ plane bisects the line segment joining the points $(1, 2, 3)$ and $(-3, 4, 5)$ at right angles. Then this plane also passes through the point

If the planes $2x - 5y + z = 8$ and $2\lambda x - 15y + \lambda z + 6 = 0$ are parallel to each other,then the value of $\lambda$ is:

What is the equation of the plane passing through the intersection of the planes $x + 2y + 3z - 4 = 0$ and $4x + 3y + 2z + 1 = 0$,and also passing through the origin?

Vedclass Test Series

Exam Paper Generator

Online Exam Module