What single force would have the same effect as two people applying a force to an object?

Arrange the two vectors tip to tail.

The resultant vector is the third side of the triangle.

Its direction is from the start (tail) of the first vector to the tip of the second vector.

Arrange the two vectors so they share a common vertex (tail to tail).

Complete the parallelogram.

The resultant vector, \({\color{BrickRed}\vec{u}}+{\color{NavyBlue}\vec{v}}\), is the diagonal drawn from the common vertex.

Notice that \({\color{BrickRed}\vec{u}}+{\color{NavyBlue}\vec{v}}={\color{NavyBlue}\vec{v}}+{\color{BrickRed}\vec{u}}\).

When does \(\lvert {\color{Violet}\vec{u}+\vec{v}}\rvert = \lvert {\color{BrickRed}\vec{u}}\rvert +\lvert {\color{NavyBlue}\vec{v}}\rvert\)?

When does \(\lvert {\color{Violet}\vec{u}+\vec{v}}\rvert = \lvert {\color{BrickRed}\vec{u}}\rvert +\lvert {\color{NavyBlue}\vec{v}}\rvert\)?

Solution

We know the shortest distance between any two points, \(A\) and \(B\), is a straight line.

Consider the case where \(\lvert {\color{Violet}\vec{u}+\vec{v}}\rvert \lt \lvert {\color{BrickRed}\vec{u}} \rvert + \lvert {\color{NavyBlue}\vec{v}} \rvert\).

We observe that for equality to hold, \({\color{BrickRed}\vec{u}}\) and \({\color{NavyBlue}\vec{v}}\) must be collinear and in the same direction.

Thus for any two collinear vectors in the same direction, \(\lvert {\color{Violet}\vec{u}+\vec{v}}\rvert = \lvert {\color{BrickRed}\vec{u}}\rvert + \lvert {\color{NavyBlue}\vec{v}}\rvert\).

The triangle law was useful when arranging vectors tip to tail.

The parallelogram law was useful when arranging vectors tail to tail.

To subtract vectors, we add the opposite.

Think

Consider the parallelogram formed by the vectors \(\vec{u}\) and \(\vec{v}\).

The sum \(\vec{u}+\vec{v}\) is the vector created by the diagonal from the tail of the two vectors \(\vec{u}+\vec{v} = \overrightarrow{AC}\).

Where is \(\vec{u}-\vec{v}\)?

We see \(\vec{u}-\vec{v}\) is the other diagonal, \(\overrightarrow{DB}\).