The Cartesian equation of the plane bar{r}=(h | vedclass

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

Question

(D) The given equation of the plane is in the form $\bar{r} = \bar{a} + \lambda \bar{A} + \mu \bar{B}$,where $\bar{a} = \hat{i} - \hat{j}$,$\bar{A} =

Vedclass · Accepted Answer

$5 x-2 y-3 z-7=0$

Vedclass · Answer

(D) The given equation of the plane is in the form $\bar{r} = \bar{a} + \lambda \bar{A} + \mu \bar{B}$,where $\bar{a} = \hat{i} - \hat{j}$,$\bar{A} = \hat{i} + \hat{j} + \hat{k}$,and $\bar{B} = \hat{i} - 2\hat{j} + 3\hat{k}$.The normal vector $\bar{n}$ to the plane is given by the cross product of the two vectors $\bar{A}$ and $\bar{B}$:$\bar{n} = \bar{A} 	imes \bar{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 1 & 1 \ 1 & -2 & 3 \end{vmatrix}$$= \hat{i}(3 - (-2)) - \hat{j}(3 - 1) + \hat{k}(-2 - 1)$$= 5\hat{i} - 2\hat{j} - 3\hat{k}$.The Cartesian equation of the plane is given by $(\bar{r} - \bar{a}) \cdot \bar{n} = 0$,which is $\bar{r} \cdot \bar{n} = \bar{a} \cdot \bar{n}$.Calculating $\bar{a} \cdot \bar{n} = (\hat{i} - \hat{j}) \cdot (5\hat{i} - 2\hat{j} - 3\hat{k}) = (1)(5) + (-1)(-2) + (0)(-3) = 5 + 2 = 7$.Thus,the Cartesian equation is $5x - 2y - 3z = 7$,or $5x - 2y - 3z - 7 = 0$.

The Cartesian equation of the plane $\bar{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2 \hat{j}+3 \hat{k})$ is

Similar Questions

If the plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{6}=1$ cuts the coordinate axes at points $A, B, C$ respectively,then the area of the triangle $ABC$ is

If the equation of a plane passing through $A(1, p, 2)$ and $B(3, 2, 4)$ and parallel to the $z$-axis is $3x - 2y - q = 0$, then:

The equation of the plane passing through $3 \hat{i}+2 \hat{j}+6 \hat{k}$ and parallel to the vectors $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ is

The equation of the plane passing through $(1,0,0)$ and $(0,1,0)$ and making an angle of $45^{\circ}$ with the plane $x+y-3=0$ is:

$A$ point on the plane determined by the points $A(1,1,-1)$, $B(2,-1,0)$, and $C(-1,0,2)$ among the following is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module