Question

The shortest distance from the point $(-10, 10, -10)$ to the $z$-axis, is

• A. $2\sqrt{10}$
• B. $10\sqrt{3}$
• C. 10
• D. $3\sqrt{10}$
• E. $10\sqrt{2}$

Hint:

Distance to $x$-axis: $\sqrt{y^2 + z^2}$.
Distance to $y$-axis: $\sqrt{x^2 + z^2}$.
Distance to $z$-axis: $\sqrt{x^2 + y^2}$.
Just omit the coordinate of the axis you are measuring to!

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The distance of a point $P(x, y, z)$ from any coordinate axis is given by the Pythagorean distance of its other two coordinates. For the $z$-axis, the foot of the perpendicular from $(x, y, z)$ is $(0, 0, z)$.
Key Formula or Approach:
Distance from point $(x, y, z)$ to the $z$-axis is $d = \sqrt{x^2 + y^2}$.

Step 2: Detailed Explanation:
The given point is $P(-10, 10, -10)$.
The coordinates are $x = -10, y = 10, z = -10$.
The foot of the perpendicular from $P$ to the $z$-axis is $M(0, 0, -10)$.
The shortest distance is the distance $PM$:
\[ d = \sqrt{(-10 - 0)^2 + (10 - 0)^2 + (-10 - (-10))^2} \]
\[ d = \sqrt{(-10)^2 + 10^2 + 0^2} \]
\[ d = \sqrt{100 + 100} = \sqrt{200} \]
\[ d = 10\sqrt{2} \]

Step 3: Final Answer:
The shortest distance is $10\sqrt{2}$.