NDA 2024 Maths · Mathematics · Vector Algebra · medium

Let vec u and vec v be any two vectors in space. 1. |vec u·vec v|≤|vec u||vec v|. 2. |vec u+vec v|≥|vec u|+|vec v|.

Single correct option

Let

\vec u

and

\vec v

be any two vectors in space.
1.

|\vec u\cdot\vec v|\le|\vec u||\vec v|

.
2.

|\vec u+\vec v|\ge|\vec u|+|\vec v|

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (A) 1 only

Statement 1 is the Cauchy-Schwarz inequality, because

|\vec u\cdot\vec v|=|\vec u||\vec v||\cos\theta|\le|\vec u||\vec v|

; it is true. Statement 2 reverses the triangle inequality, which is

|\vec u+\vec v|\le|\vec u|+|\vec v|

, with equality only when the vectors point in the same direction; for

\hat i

and

\hat j

\sqrt2<2

, so statement 2 is false.

Original question written in the NDA 2024 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake