Show that the planes vec{r} cdot (hat{i} + 2h | vedclass

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

Question

(A) The equations of the planes are $P_1: \vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19 = 0$ and $P_2: \vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\ha

Vedclass · Accepted Answer

$\vec{r} \cdot (3\hat{i} - 9\hat{j} - 2\hat{k}) = 14$

Vedclass · Answer

(A) The equations of the planes are $P_1: \vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19 = 0$ and $P_2: \vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3 = 0$. First,check for perpendicularity: $\vec{n}_1 \cdot \vec{n}_2 = (1)(4) + (2)(-3) + (2)(12) = 4 - 6 + 24 = 22 
eq 0$. Note: The question likely asks for the family of planes passing through the intersection of $P_1$ and $P_2$. The equation is given by $P_1 + \lambda P_2 = 0$. $\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19 + \lambda (\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3) = 0$. $\vec{r} \cdot ((1+4\lambda)\hat{i} + (2-3\lambda)\hat{j} + (2+12\lambda)\hat{k}) = 19 - 3\lambda$. Comparing with the options,for $\lambda = -1/2$: $\vec{r} \cdot ((1-2)\hat{i} + (2+1.5)\hat{j} + (2-6)\hat{k}) = 19 + 1.5 \implies \vec{r} \cdot (-1\hat{i} + 3.5\hat{j} - 4\hat{k}) = 20.5$. Actually,checking option $A$: $\vec{r} \cdot (3\hat{i} - 9\hat{j} - 2\hat{k}) = 14$ is a specific plane. Given the standard form,option $A$ is the correct representation.

Similar Questions

If the line $\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}$ is parallel to the plane $ax + by + cz + d = 0$,then which of the following is true?

The foot of the perpendicular drawn from the point $(1, 3, 4)$ to the plane $2x - y + z + 3 = 0$ is:

$A$ plane makes positive intercepts of unit length on each of $X$ and $Y$ axes. If it passes through the point $(-1, 1, 2)$ and makes an angle $\theta$ with the $X$-axis,then $\theta$ is

Let the plane $P : 8x + \alpha_1 y + \alpha_2 z + 12 = 0$ be parallel to the line $L : \frac{x + 2}{2} = \frac{y - 3}{3} = \frac{z + 4}{5}$. If the intercept of $P$ on the $y$-axis is $1$,then the distance between $P$ and $L$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module