Q.The distance of the point with position vector 3𝑖̂ + 4𝑗̂ + 5𝑘̂ from the y-axis is
(A) 4 units
(B) √34 units
(C) 5 units
(D) 5√2 units

Concept: Distance from the y-axis means the perpendicular distance to the y-axis, which is the magnitude of the projection of the point onto the xz-plane (i.e., ignoring the y-coordinate).

Steps:

1. The given point is $(3, 4, 5)$.
2. Distance from the y-axis depends only on the $x$ and $z$ coordinates: $\sqrt{x^2 + z^2}$. …

Why distance from the y-axis?

When we ask for the distance of a point from the y-axis, we mean the shortest distance between the point and any point on the y-axis. The y-axis is the set of all points where $x = 0$ and $z = 0$ — only the y-coordinate varies. So the perpendicular from our point to the y-axis will land at $(0,4,0)$, because the y-coordinate stays the same (the foot of the perpendicular shares the same y-value).

This is exactly like finding the distance of a point $(x,y)$ from the y-axis in 2D: you drop the y-coordinate and take $|x|$. In 3D, the y-axis is a line, so the distance is the length of the component perpendicular to it — which lives entirely in the xz-plane.

The y-coordinate plays no role because moving along the y-axis doesn't change the perpendicular distance.

Step-by-step solution

1. Identify the coordinates.

   The position vector $3\hat{i} + 4\hat{j} + 5\hat{k}$ corresponds to the point $(3,4,5)$.

2. Visualise the geometry. …

Method: Distance of a point from a coordinate axis

Use this to find the perpendicular distance from a point to one of the coordinate axes in 3D.

Steps

Step 1: Identify which axis you are measuring from.

The axis is a line, so the shortest distance is the perpendicular dropped onto it. The coordinate measured ALONG that axis does not affect the distance.

Step 2: Drop the coordinate of that axis and combine the other two.

$\text{from }x\text{-axis}=\sqrt{y^2+z^2},\quad \text{from }y\text{-axis}=\sqrt{x^2+z^2},\quad \text{from }z\text{-axis}=\sqrt{x^2+y^2}.$ …

Common Mistakes

Mistake 1: Including the $y$-coordinate in the distance.

Why it's wrong: $\sqrt{3^2+4^2+5^2}=5\sqrt2$ is the distance from the ORIGIN, not from the $y$-axis. Correct approach: drop the $y$-coordinate and use $\sqrt{x^2+z^2}=\sqrt{34}$.

Mistake 2: Using the 2D rule $|x|$ and answering $3$. …