- Finding horizontal and vertical components of a force
- Combining two or more forces
- Forces in equilibrium
- Finding a missing force given the resultant force
Part 1: Finding horizontal and vertical components of a force
A matrix is an array of elements.
Part 2: Combining two or more forces
You can find the resultant, or net effect, of two or more forces (given as vectors) by adding these vectors. Here are a couple of simple illustrations:
- If someone exerts a pulling force of 5 N to the right on an object, and someone else exerts a pulling force of 5N to the left, the resultant force on the object is 0 N.
- If someone exerts a pulling force of 5 N to the right on an object, and someone else exerts a pulling force of 4N to the left, the resultant force on the object is 1 N to the right.
Adding vectors
Before getting into the detail of multiplying a matrix by another matrix, we’ll take a look at a simple situation to help illustrate the principle behind matrix multiplication:
Considering components in perpendicular directions
We can use index notation with matrices to indicate repeated multiplication. As you might expect:
- A2 = A \( \times \) A
- A3 = A \( \times \)A \( \times \) A
Part 3: Forces in equilibrium
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Part 4: Finding a missing force given the resultant force
A 2 × 2 matrix can be used to apply a linear transformation to points on a Cartesian grid. A linear transformation in two dimensions has the following properties:
- The origin (0,0) is mapped to the origin (it is invariant) under the transformation
- Straight lines are mapped to straight lines under the transformation
- Parallel lines remain parallel under the transformation