Points $A$ and $B$ have coordinates $\left( 1,2,1 \right)$ and $\left( 1,-5,2 \right)$ respectively.

Question

Points $A$ and $B$ have coordinates $\left( 1,2,1 ight)$ and $\left( 1,-5,2 ight)$ respectively. The line $BC$ has equation $\mathbf{r}=\left( \begin{matrix}1 \-5 \2 \\end{matrix} ight)+\lambda \left( \begin{matrix}-2 \1 \1 \\end{matrix} ight)$, $\lambda \in \mathbb{R}$. Given that $AB=AC$, show that the coordinates of point $C$ is $\left( -3,-3,4 ight)$. [3] Find the cosine of angle $BAC$ and hence find the exact area of triangle $ABC$. [4] Plane $p$ is parallel to plane $ABC$ and it is $\sqrt{\frac{11}{6}}$ units from plane $ABC$. Find the possible equations of $p$. [4] Point $D$ lies on $p$. Deduce the exact volume of the tetrahedron $ABCD$. [2] [The volume of a tetrahedron of base area $A$ and height $h$ is given by $V=\frac{1}{3}Ah$.]

Accepted Answer

(i) $(-3, -3, 4)$ (ii) $2\sqrt{66}$ units² (iii) $\mathbf{r} \cdot \begin{pmatrix} 4 \\ 1 \\ 7 \end{pmatrix} = 24$ and $\mathbf{r} \cdot \begin{pmatrix} 4 \\ 1 \\ 7 \end{pmatrix} = 2$ (iv) $\frac{22}{3}$ units³

2020 TJC Promo Q10

2020 TJC Promo Q10

Video Solution:

Solution:

Solution locked