First slide

Introduction to 3-D Geometry

Question

If $A (1, 2, 3), B (2, 3, 4)$ and $A B$ is produced up to $C$ such that $2 A B = B C$ , then $C =$

Easy

A

$(5, 4, 6)$
B

$(4, 1, 4)$
C

$(\frac{5}{2}, \frac{7}{2}, \frac{9}{2})$
D

$(4, 5, 6)$

Solution

Given $2 A B = B C$
it gives
$\begin{matrix} \bar{A B} = \bar{B C} \\ 2 (\bar{O B} - \bar{O A}) = (\bar{O C} - \bar{O B}) \\ 2 \bar{O B} - 2 \bar{O A} + \bar{O B} = \bar{O C} \\ \bar{O C} = 3 \bar{O B} - 2 \bar{O A} \end{matrix}$
It implies
$\begin{matrix} \bar{O C} = 3 (2 i + 3 j + 4 k) - 2 (i + 2 j + 3 k) \\ = 6 i + 9 i + 12 k - 2 i - 4 j - 6 k \\ = 4 i + 5 j + 6 k \end{matrix}$
Therefore, $C (4, 5, 6)$

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