Question

The coordinates of the point $P \equiv (1,2,3)$ and $O \equiv (0,0,0)$ are given. The direction cosines of $\overline{OP}$ are

• A. $\frac{1}{\sqrt{14}},\ \frac{2}{\sqrt{14}},\ \frac{3}{\sqrt{14}}$
• B. $\frac{1}{\sqrt{6}},\ \frac{2}{\sqrt{6}},\ \frac{1}{\sqrt{6}}$
• C. $\frac{1}{\sqrt{3}},\ \frac{1}{\sqrt{3}},\ \frac{1}{\sqrt{3}}$
• D. $\frac{2}{\sqrt{29}},\ \frac{3}{\sqrt{29}},\ \frac{4}{\sqrt{29}}$

Hint:

The sum of squares of direction cosines must always equal exactly 1 ($\ell^2 + m^2 + n^2 = 1$).
Let's verify option (A): $\left(\frac{1}{\sqrt{14}}\right)^2 + \left(\frac{2}{\sqrt{14}}\right)^2 + \left(\frac{3}{\sqrt{14}}\right)^2 = \frac{1}{14} + \frac{4}{14} + \frac{9}{14} = \frac{14}{14} = 1$. This mathematical property lets you filter out invalid combinations instantly!

Answer 1

The Correct Option is A

Step 1: Read the points.
We have $O(0,0,0)$ and $P(1,2,3)$. We want the direction cosines of the segment $\overline{OP}$.
Step 2: Find the direction ratios.
For a line from the origin to $P(x,y,z)$, the direction ratios are just $(x,y,z) = (1,2,3)$.
Step 3: Recall direction cosines.
Divide each direction ratio by the length $r = \sqrt{x^2 + y^2 + z^2}$.
Step 4: Find the length.
\[ r = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]
Step 5: Divide each ratio.
\[ \ell = \frac{1}{\sqrt{14}},\quad m = \frac{2}{\sqrt{14}},\quad n = \frac{3}{\sqrt{14}} \]
Step 6: State the answer.
These are the direction cosines of $\overline{OP}$. \[ \boxed{\frac{1}{\sqrt{14}},\ \frac{2}{\sqrt{14}},\ \frac{3}{\sqrt{14}}} \]