Class 11 NCERT PHYSICS • Chapter 3: Motion in a Plane • Question 7

QUESTION 7

Given a + b + c + d = 0, which of the following statements are correct :

(a) a, b, c, and d must each be a null vector,

(b) The magnitude of (a + c) equals the magnitude of ( b + d),

(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,

(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear

SOLUTION

We need to analyze the properties of four vectors whose resultant sum is zero and identify the correct statements.

Concept: Polygon Law of Vector Addition

The Polygon Law of Vector Addition states that if a number of vectors are represented by the sides of a closed polygon taken in order, their resultant is zero. Key properties include:

Triangle Inequality: $|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$ .
Coplanarity: The sum of two vectors lies in the plane defined by those two vectors.

Given:

$\vec{a} + \vec{b} + \vec{c} + \vec{d} = \vec{0}$

Solving:

Analysis of statement (a):
- For the sum $\vec{a} + \vec{b} + \vec{c} + \vec{d} = \vec{0}$ , it is not necessary for each vector to be a null vector. They can be non-zero vectors that form a closed quadrilateral.
- Therefore, statement (a) is incorrect.
Analysis of statement (b):
- From the given equation, we can rearrange the terms:

\vec{a} + \vec{c} = -(\vec{b} + \vec{d})

Taking the magnitude of both sides:

|\vec{a} + \vec{c}| = |-(\vec{b} + \vec{d})| = |\vec{b} + \vec{d}|

Therefore, statement (b) is correct.

Analysis of statement (c):
- We can write $\vec{a}$ in terms of the other vectors:

\vec{a} = -(\vec{b} + \vec{c} + \vec{d})

Using the generalized triangle inequality:

|\vec{a}| = |\vec{b} + \vec{c} + \vec{d}| \le |\vec{b}| + |\vec{c}| + |\vec{d}|

This shows the magnitude of $\vec{a}$ can never exceed the sum of the magnitudes of the other three.
Therefore, statement (c) is correct.

Analysis of statement (d):
- Rearranging the given equation:

\vec{b} + \vec{c} = -(\vec{a} + \vec{d})

The vector $(\vec{a} + \vec{d})$ must lie in the plane containing $\vec{a}$ and $\vec{d}$ (if they are not collinear). Consequently, its negative, $-(\vec{a} + \vec{d})$ , which equals $\vec{b} + \vec{c}$ , must also lie in that same plane.
If $\vec{a}$ and $\vec{d}$ are collinear, their sum and its negative lie along the same line.
Therefore, statement (d) is correct.

Answer:

\text{(b), (c), and (d) are correct}