First slide

Direction ratios and direction cosines

Question

If $(2, 3, 4)$ and $(1, 2, 3)$ are direction ratios of $\vec{O A}$ and $\vec{O B}$ , then the direction cosines of a line perpendicular to both the lines $\vec{O A}$ and $\vec{O B}$ are

Moderate

A

$(\frac{1}{\sqrt{6}}, \frac{- 2}{\sqrt{6}}, \frac{1}{\sqrt{6}})$
B

$(\frac{1}{\sqrt{6}}, \frac{- 2}{\sqrt{6}}, \frac{- 1}{\sqrt{6}})$
C

$(\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{- 1}{\sqrt{6}})$
D

None

Solution

The direction ratio of any line perpendicular to both lines $O A, O B$ is along the vector $\bar{O A} \times \bar{O B}$
It implies,
$\begin{matrix} \bar{O A} \times \bar{O B} = |\begin{matrix} i & j & k \\ 2 & 3 & 4 \\ 1 & 2 & 3 \end{matrix}| \\ = i (9 - 8) - j (6 - 4) + k (4 - 3) \\ = i - 2 j + k \end{matrix}$
Hence, the direction ratios of the line perpendicular to the lines are $〈1, - 2, 1〉$
Therefore, the required direction cosines are $〈\frac{1}{\sqrt{6}}, - \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}〉$

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