Let vec{a}, vec{b}, vec{c} be three non-copla | vedclass

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

Question

(B) The line $L$ passes through $P_1 = \vec{a}-\vec{b}+\vec{c}$ and $P_2 = \vec{b}-\vec{c}$. The direction vector of $L$ is $\vec{v} = P_2 - P_1 = (\v

Vedclass · Accepted Answer

$\vec{b}+\vec{c}$

Vedclass · Answer

(B) The line $L$ passes through $P_1 = \vec{a}-\vec{b}+\vec{c}$ and $P_2 = \vec{b}-\vec{c}$. The direction vector of $L$ is $\vec{v} = P_2 - P_1 = (\vec{b}-\vec{c}) - (\vec{a}-\vec{b}+\vec{c}) = -\vec{a} + 2\vec{b} - 2\vec{c}$.Thus, the equation of line $L$ is $\vec{r} = (\vec{a}-\vec{b}+\vec{c}) + \lambda(-\vec{a} + 2\vec{b} - 2\vec{c})$.The plane $\pi$ passes through $A = 2\vec{a}-\vec{b}$, $B = 2\vec{b}-\vec{c}$, and $C = 2\vec{c}-\vec{a}$.The normal vector to the plane is $\vec{n} = (B-A) 	imes (C-A) = (-2\vec{a}+3\vec{b}-\vec{c}) 	imes (-3\vec{a}+\vec{b}+2\vec{c}) = 7(\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c} + \vec{c} 	imes \vec{a})$.The equation of the plane is $(\vec{r} - (2\vec{a}-\vec{b})) \cdot (\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c} + \vec{c} 	imes \vec{a}) = 0$.Substituting $\vec{r} = \vec{a}-\vec{b}+\vec{c} + \lambda(-\vec{a} + 2\vec{b} - 2\vec{c})$ into the plane equation and solving for $\lambda$, we find $\lambda = 1$.Substituting $\lambda = 1$ into the line equation gives $\vec{r} = (\vec{a}-\vec{b}+\vec{c}) + 1(-\vec{a} + 2\vec{b} - 2\vec{c}) = \vec{b}-\vec{c}$.

Similar Questions

The equation of the line passing through $(-4, 1, 3)$,parallel to the plane $x + 2y - z - 5 = 0$ and intersecting the line $\frac{x + 1}{-3} = \frac{y - 3}{2} = \frac{z - 2}{-1}$ is

The plane $lx + my = 0$ is rotated by an angle $\alpha$ about its line of intersection with the plane $z = 0$. Find the equation of the plane in its new position.

For what value of $c$ does the line $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{-1}$ intersect the curve $xy = c^2, z = 0$?

Let the image of the point $P(1, 2, 3)$ in the plane $2x - y + z = 9$ be $Q$. If the coordinates of the point $R$ are $(6, 10, 7)$,then the square of the area of the triangle $PQR$ is $.....$.

What is the length of the perpendicular drawn from the origin to the plane passing through point $a$ and containing the line $r = b + \lambda c$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module