The vector sum of two forces is perpendicular to their vector differences. In that case the forces

Question

Given vectors \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear and \((\vec{a}+\vec{b})\) is collinear with \((\vec{b}+\vec{c})\) which is collinear with \(\vec{a}\), and \(|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{2}\), find \(|\vec{a}+\vec{b}+\vec{c}|\).

Answer 1

To solve the given problem, let's first understand the concepts involved with vector operations:

We have two vectors representing forces, say \(\mathbf{A}\) and \(\mathbf{B}\). The vector sum is given by:

\(\mathbf{A} + \mathbf{B}\)

The vector difference is given by:

\(\mathbf{A} - \mathbf{B}\)

According to the problem, the vector sum is perpendicular to the vector difference. Two vectors are perpendicular if their dot product is zero. Therefore, we have:

(\mathbf{A} + \mathbf{B}) \cdot (\mathbf{A} - \mathbf{B}) = 0

Expanding the dot product, we get:

\((\mathbf{A} \cdot \mathbf{A}) - (\mathbf{A} \cdot \mathbf{B}) + (\mathbf{B} \cdot \mathbf{A}) - (\mathbf{B} \cdot \mathbf{B}) = 0\)

Simplifying the above expression,

\(\mathbf{A} \cdot \mathbf{A} - \mathbf{B} \cdot \mathbf{B} = 0\)

This implies:

|\mathbf{A}|^2 = |\mathbf{B}|^2, which means \(|\mathbf{A}| = |\mathbf{B}|\)

Thus, the magnitudes of the two vectors are equal, but not necessarily their directions. Hence, the correct answer is:

are equal to each other in magnitude

This result confirms that the forces have the same magnitude, satisfying the condition given in the problem.

Answer 2