---
title: "In a region of space, the three-dimensional electric potential is given by \\(V(x,y,z) = Cxyz\\), where \\(C\\) is a positive constant. Which of the following correctly identifies the electric field vector \\(\\vec{E}(x,y,z)\\) and the set of points where \\(\\vec{E}\\) is perpendicular to the position vector \\(\\vec{r} = x\\hat{i} + y\\hat{j} + z\\hat{k}\\)?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/118123/"
date_modified: "2026-08-04T08:05:17+00:00"
---

# In a region of space, the three-dimensional electric potential is given by \(V(x,y,z) = Cxyz\), where \(C\) is a positive constant. Which of the following correctly identifies the electric field vector \(\vec{E}(x,y,z)\) and the set of points where \(\vec{E}\) is perpendicular to the position vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\)?

In a region of space, the three-dimensional electric potential is given by \(V(x,y,z) = Cxyz\), where \(C\) is a positive constant. Which of the following correctly identifies the electric field vector \(\vec{E}(x,y,z)\) and the set of points where \(\vec{E}\) is perpendicular to the position vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\)?

- **A.** Field: \(\vec{E} = C(yz\hat{i} + xz\hat{j} + xy\hat{k})\); Perpendicular locations: points along the line \(x = y = z\)
- **B.** Field: \(\vec{E} = -C(yz\hat{i} + xz\hat{j} + xy\hat{k})\); Perpendicular locations: points along the line \(x = y = z\)
- **C.** Field: \(\vec{E} = -C(yz\hat{i} + xz\hat{j} + xy\hat{k})\); Perpendicular locations: points on the coordinate planes \(x = 0\), \(y = 0\), or \(z = 0\)
- **D.** Field: \(\vec{E} = -C(x\hat{i} + y\hat{j} + z\hat{k})\); Perpendicular locations: points on any sphere centered at the origin

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118123/*