The equations of three planes ${{p}_{1}}$, ${{p}_{2}}$ and ${{p}_{3}}$ are \(\mathbf{r}\cdot \

Question

The equations of three planes ${{p}_{1}}$, ${{p}_{2}}$ and ${{p}_{3}}$ are $\mathbf{r}\cdot \left( \begin{matrix} 3 \ 2 \ -5 \ \end{matrix} \right)=3$, $\mathbf{r}\cdot \left( \begin{matrix} -5 \ 3 \ 2 \ \end{matrix} \right)=-5$ and $\mathbf{r}\cdot \left( \begin{matrix} 17 \ 5 \ a \ \end{matrix} \right)=b$ respectively, where $a$ and $b$ are constants. If $a=-20.9$ and $b=16.6$, find the coordinates of the point at which these planes meet.[2] The planes ${{p}_{1}}$ and ${{p}_{2}}$ intersect in a line $l$. Find a vector equation of $l$.[2] Given that all three planes meet in the line $l$, find the values of $a$ and $b$.[3]

Accepted Answer

The planes meet at $\left(\dfrac7{11},-\dfrac4{11},-\dfrac4{11}\right)$. (i) $l:\mathbf r=\begin{pmatrix}1\0\0\end{pmatrix}+\lambda\begin{pmatrix}1\1\1\end{pmatrix}$, $\lambda\in\mathbb R$ (ii) $a=-22$, $b=17$

2023 NJC Promo Q1

2023 NJC Promo Q1

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