If the line joining the points hat{i}+hat{j} | vedclass

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

Question

(B) The line passes through $A(1, 1, 0)$ and $B(3, 1, -1)$. The direction vector is $\vec{v} = B - A = 2\hat{i} + 0\hat{j} - 1\hat{k}$. The equation o

Vedclass · Accepted Answer

$29 \hat{i}+\hat{j}-14 \hat{k}$

Vedclass · Answer

(B) The line passes through $A(1, 1, 0)$ and $B(3, 1, -1)$. The direction vector is $\vec{v} = B - A = 2\hat{i} + 0\hat{j} - 1\hat{k}$. The equation of the line is $\frac{x-1}{2} = \frac{y-1}{0} = \frac{z}{-1} = r$. Thus,any point $P$ on the line is $(2r+1, 1, -r)$.The plane passes through $(2, 4, 0)$ and is parallel to $\vec{u} = 3\hat{j} + 5\hat{k}$ and $\vec{w} = 3\hat{i} - \hat{k}$. The normal to the plane is $\vec{n} = \vec{u} 	imes \vec{w} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 3 & 5 \ 3 & 0 & -1 \end{vmatrix} = -3\hat{i} + 15\hat{j} - 9\hat{k}$.The equation of the plane is $-3(x-2) + 15(y-4) - 9(z-0) = 0$,which simplifies to $x - 5y + 3z + 18 = 0$.Substituting $P(2r+1, 1, -r)$ into the plane equation: $(2r+1) - 5(1) + 3(-r) + 18 = 0 \Rightarrow -r + 14 = 0 \Rightarrow r = 14$.The position vector of $P$ is $(2(14)+1)\hat{i} + 1\hat{j} - 14\hat{k} = 29\hat{i} + \hat{j} - 14\hat{k}$.

If the line joining the points $\hat{i}+\hat{j}$ and $3 \hat{i}+\hat{j}-\hat{k}$ meets the plane that passes through the point $2 \hat{i}+4 \hat{j}$ and is parallel to the vectors $3 \hat{j}+5 \hat{k}$ and $3 \hat{i}-\hat{k}$ at point $P$,then the position vector of the point $P$ is

Similar Questions

The equation of the plane passing through the line of intersection of planes $\pi_1: 2x + 6y + 4z - 7 = 0$ and $\pi_2: x - y - 2z - 2 = 0$, and perpendicular to the plane $x + y + 2z - 5 = 0$ is:

If the line passing through the points $(a, 2, -4)$ and $(5, 3, b)$ crosses the $ZX$-plane at the point $(-a+2b, 0, a+b)$, then find the value of $14a+7b$.

$A$ line with direction cosines passes through the point $P(2, -1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x + y + z = 9$ at point $Q$. Find the length of the line segment $PQ$.

If the lines $\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}$ and $\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}$ are coplanar,then the equation of the plane containing these lines is:

If $L$ is the line of intersection of two planes $x+2y+2z=15$ and $x-y+z=4$ and the direction ratios of the line $L$ are $(a, b, c)$, then $\frac{a^2+b^2+c^2}{b^2}=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module