June Revision Exercise 8 Q6

(a)
$\vec{OC}$

$= \frac{\vec{OA}+3\vec{OB}}{4}$

$= \frac{1}{4} [{\begin{pmatrix}1 \\ 1 \\ 2 \end{pmatrix} + \begin{pmatrix}3 \\ 5 \\ 6 \end{pmatrix}}]$

$= \begin{pmatrix}2.5 \\ 4 \\ 5 \end{pmatrix}$

$X \text{lies in } \pi: \begin{pmatrix}2.5 \\ 4 \\ 5 \end{pmatrix} \bullet \begin{pmatrix}2 \\ \lambda \\ 0 \end{pmatrix}= \mu$

$5 + 4 \lambda = \mu$

(b)
$\vec{AB} = \begin{pmatrix}3 \\ 5 \\ 6 \end{pmatrix} - \begin{pmatrix}1 \\ 1 \\ 2 \end{pmatrix} = \begin{pmatrix}2 \\ 4 \\ 4 \end{pmatrix}$

If the plane $\pi$ contains the line $AB$ , the $p_3$ is parallel to $AB$ :

$\begin{pmatrix}2 \\ 4 \\ 4 \end{pmatrix} \bullet \begin{pmatrix}2 \\ \lambda \\ 0 \end{pmatrix} = 0$

$4 + 4 \lambda = 0$

$\lambda = -1$

$\lambda = -1$ and the point $A$ lies in the plane $p_3$ :

$\begin{pmatrix}1 \\ 1 \\ 2 \end{pmatrix} \bullet \begin{pmatrix}2 \\ -1\\ 0 \end{pmatrix} = \mu$

$2 - 1 = \mu$

$\mu = 1$

Alternatively, you may consider that $A \text{and} B \text{line in the plane} p_3$ :

$\begin{pmatrix}1 \\ 1 \\ 2 \end{pmatrix} \bullet \begin{pmatrix}2 \\ \lambda \\ 0 \end{pmatrix} = \mu$

$\Rightarrow 2 + \lambda = \mu$ — (1)

$\begin{pmatrix}3 \\ 5 \\ 6 \end{pmatrix} \bullet \begin{pmatrix}2 \\ \lambda \\ 0 \end{pmatrix} = \mu$

$\Rightarrow 6 + 5\lambda = \mu$ — (2)

Solving, $\lambda = -1, \mu = 1$

Back to June Revision Exercise 8.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

Comments

pingbacks / trackbacks

June Revision Set 8 – The Culture
[…] June Revision Exercise 8 Q6 – The Culture on June Revision Set 8 […]
June 10th, 2016 09:28 PM

Cart

June Revision Exercise 8 Q6

Leave a Reply to June Revision Set 8 – The Culture Cancel reply