Question

For non – zero vectors $\vec{a}$ and $\vec{b}$ if $|\vec{a} + \vec{b} | < |\vec{a} - \vec{b} | $, then $\vec{a}$ and $\vec{b}$ are
a) Collinear
b) Perpendicular to each other
c) Inclined at an acute angle
d) Inclined at an obtuse angle

Answer 1

Hint: If a vector is non-zero, it has at least one non-zero component; a zero vector has all parts as zero, therefore, no length. A non-zero vector is one in which at least one non-zero is there, at least in absolute numbers. In general, a non-zero vector is not the identity element for the summation of the vector space.

Complete step-by-step solution:
Given: $\vec{a}$ and $\vec{b}$ are non – zero vectors.
$|\vec{a} + \vec{b} | < |\vec{a}- \vec{b} | $.
Squaring both sides.
$a^{2} + b^{2} + 2ab\ cos \theta < a^{2} + b^{2} - 2ab\ cos \theta$
$2ab\ cos \theta <-2ab\ cos \theta $
$4ab\ cos \theta < 0 $
$\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
$\vec{a}$ and $\vec{b}$ are non – zero vectors.
$\vec{a}$ and $\vec{b}$ are Positive.
$a, b >0$
$\therefore cos\theta <0$
Cosine function is negative in the second and third quadrant.
So, $\vec{a}$ and $\vec{b}$ are inclined at an obtuse angle.
Option (d) is correct.

Note:If the sum of two non-zero vectors is equal to their difference, then since the angle between given vectors is $90^{\circ}$, The vectors are perpendicular. The main difference between unit vector and non-zero vector is that the unit vector is the outcome of normalizing a non-zero vector and the unit vector is the ratio of vector to its length.