Question

The projections of a line segment on the coordinate axes are \(5,6,8\). Then the length of the line segment is

• A. \(5\)
• B. \(5\sqrt{5}\)
• C. \(6\)
• D. \(6\sqrt{6}\)
• E. \(6\sqrt{5}\)

Hint:

The length of a 3D line segment can be found using the formula \(\sqrt{x^2+y^2+z^2}\), where \(x,y,z\) are projections on axes.

Answer 1

The Correct Option is B

Step 1: Understand the concept of projections.
The projections of a line segment on the coordinate axes represent the components of the vector along \(x\), \(y\), and \(z\) axes.

Step 2: Represent the projections as components.
Thus, the components of the line segment can be taken as:
\[ (5,\ 6,\ 8) \]

Step 3: Recall the formula for length of a vector.
If a vector has components \((a,b,c)\), then its magnitude is:
\[ \sqrt{a^2+b^2+c^2} \]

Step 4: Apply the formula.
\[ \text{Length}=\sqrt{5^2+6^2+8^2} \]

Step 5: Simplify the expression.
\[ = \sqrt{25+36+64} \] \[ = \sqrt{125} \]

Step 6: Simplify further.
\[ \sqrt{125}=\sqrt{25\cdot 5}=5\sqrt{5} \]

Step 7: State the final answer.
Thus, the length of the line segment is:
\[ \boxed{5\sqrt{5}} \] which matches option \((2)\).