---
title: "Condition for constant magnitude of vector"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:condition-constant-magnitude-vector"
source: https://www.bctengineer.com/q/semester_2/mathematics/condition-constant-magnitude-vector
site: BCT Engineering
---

# Condition for constant magnitude of vector

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2074 Bhadra, 5 marks; 2070 Bhadra, 5 marks; 2069 Bhadra, 5 marks; 2067 Mangsir, 5 marks; 2067 Chaitra, 5 marks (asked 5 times)
- Video: [The Necessary and sufficient condition forthe vector a(t) to have a constant Magnitude is a. da/dt=0](https://www.youtube.com/watch?v=JI2HSfNjPCc) — Ocus Ready Coaching
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/condition-constant-magnitude-vector
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Prove that the necessary and sufficient condition for the vector function a of scalar variable 't' to have constant magnitude is a · da/dt = 0.

## Short answer

Since |a|² = a·a, its derivative is 2a·da/dt. If |a| is constant, derivative zero so a·da/dt = 0. If a·da/dt = 0, then |a|² is constant, so magnitude constant. Thus a·da/dt = 0 is necessary and sufficient for constant magnitude.

## Answer

Let a(t) be a vector function of the scalar t. Its magnitude |a| is constant exactly when |a|² = a · a is constant (|a| ≥ 0).

#### Derivative of a · a
Using the product rule for dot products:

d/dt (a · a) = (da/dt) · a + a · (da/dt) = 2 a · (da/dt).

#### Necessary condition
If |a| is constant, then a · a = |a|² is constant, so d/dt(a · a) = 0. Thus 2 a · da/dt = 0, i.e.

**a · da/dt = 0.**

#### Sufficient condition
Let a · da/dt = 0. Then d/dt(a · a) = 2a · da/dt = 0, so a · a = |a|² does not change with t. Since |a| ≥ 0, |a| is constant.

**Hence |a| is constant if and only if a · da/dt = 0.**

Geometric meaning: a vector of constant length can only turn, so its derivative is perpendicular to it (for example, the position vector of a point moving on a circle or sphere centred at the origin).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. |a| is constant exactly when which quantity is constant?
   A. a · a
   B. a × a
   C. da/dt
   D. a + da/dt
   Answer: A. |a|^2 = a · a and |a| ≥ 0. (Key point: Reduction)

2. d/dt(a · a) equals:
   A. a · da/dt
   B. 2 a × da/dt
   C. da/dt · da/dt
   D. 2 a · da/dt
   Answer: D. The dot product rule gives two equal terms. (Key point: Derivative)

3. a · da/dt = 0 means:
   A. da/dt is parallel to a
   B. a is zero
   C. da/dt is perpendicular to a
   D. a has constant direction
   Answer: C. A vector of fixed length can only turn, so its derivative is perpendicular to it. (Key point: Geometric meaning)

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