Question

Given that \(\mathbf{A}+\mathbf{B}=0\), (a) how does the magnitude of \(\overrightarrow{\mathbf{B}}\) compare with the magnitude of \(\overrightarrow{\mathbf{A}}\) ? (b) How does the direction of \(\overrightarrow{\mathbf{B}}\) compare with the direction of \(\overrightarrow{\mathbf{A}}\) ?

Short answer

(a) Equal magnitudes. (b) Opposite directions.

Step-by-step solution

Understanding the Vector Equality

We are given that \( \mathbf{A} + \mathbf{B} = 0 \). This equation tells us that vector \( \mathbf{A} \) and vector \( \mathbf{B} \) are equal in magnitude but opposite in direction such that when added, they result in the zero vector.

Analyzing Magnitudes

Since \( \mathbf{A} + \mathbf{B} = 0 \), the vectors must cancel each other out completely. This means that the magnitude of \( \overrightarrow{\mathbf{B}} \) has to be equal to the magnitude of \( \overrightarrow{\mathbf{A}} \). Mathematically, this implies \( |\overrightarrow{\mathbf{A}}| = |\overrightarrow{\mathbf{B}}| \).

Analyzing Directions

For the summation of two vectors to be zero, their directions must be opposite. Therefore, the direction of vector \( \overrightarrow{\mathbf{B}} \) is exactly opposite to that of vector \( \overrightarrow{\mathbf{A}} \).

Key concepts

Opposite Vectors

In the world of vectors, opposite vectors play a crucial role. When vectors are opposite, they cancel each other out when added together. In simple terms, if two vectors, say \( \mathbf{A} \) and \( \mathbf{B} \), are opposite, then they are equal in magnitude but point in exactly opposite directions.
Think of them like two people pushing on the same door from opposite sides with equal force – the door doesn't move.
This is why when you see an equation like \( \mathbf{A} + \mathbf{B} = 0 \), it means \( \mathbf{A} \) and \( \mathbf{B} \) are opposites. When added, they create a zero vector because their effects cancel each other out entirely.

Vector Magnitude

Vector magnitude is all about the size or length of the vector and is always a positive number.
Magnitude does not consider direction; it only tells us how much of the vector there is. In mathematical terms, if \( \overrightarrow{\mathbf{A}} = \langle a_1, a_2, a_3 \rangle \), its magnitude is calculated as follows:

• \( |\overrightarrow{\mathbf{A}}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)

In the case where \( \mathbf{A} + \mathbf{B} = 0 \), we know:

• The magnitude of \( \overrightarrow{\mathbf{B}} \) is the same as \( \overrightarrow{\mathbf{A}} \) since they cancel each other completely.
  Simply put, if the vectors' magnitudes were different, one vector would overpower the other, and the sum wouldn't be zero.

Vector Direction

Vector direction defines where the vector is pointing. It is often represented as an angle in relation to a reference axis.
When we talk about vectors being in opposite directions, like \( \mathbf{A} \) and \( \mathbf{B} \) in our original problem, it means one points exactly opposite the other.

• For example, on a plane, if vector \( \mathbf{A} \) points north, vector \( \mathbf{B} \) might point south.

This opposite direction is crucial for achieving a sum of zero. Without opposite directions, even equal magnitudes would not cancel each other out.
It's like walking five steps forward then five steps backward; your net movement (or vector sum) is zero because the directions are opposite.