The line $l$ passes through the points $A(2,-1, 3)$ and $B(4, 1, 5)$. The plane $p_1$ has eq

Question

The line $l$ passes through the points $A(2,-1, 3)$ and $B(4, 1, 5)$. The plane $p_1$ has equation $2x-y+2z=0$. Find the vector equation of the line $l$.[2] The plane $p_2$ contains $l$ and the point $C$ with coordinates $(3, 6, 0)$. Show that a vector perpendicular to $p_2$ is $\begin{pmatrix}-5\2\3\end{pmatrix}$. Hence find the cartesian equation of $p_2$. [3] Find the vector equation of the line of intersection between plane $p_1$ and plane $p_2$.[2] Find the acute angle between the planes $p_1$ and $p_2$.[2] Find the position vector of the point $F$, the foot of perpendicular from point $B$ to the plane $p_1$.[4]

Accepted Answer

$\mathbf r=(2,-1,3)+\lambda(1,1,1)$; $p_2:-5x+2y+3z=-3$; intersection $\mathbf r=(3,6,0)+\mu(7,16,1)$; $71.1^\circ$; $\overrightarrow{OF}=\dfrac19(2,26,11)$

2025 MI PU1 Promo Q10

2025 MI PU1 Promo Q10

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