To combine vectors into one vector, you typically perform vector addition. This process involves placing the vectors head-to-tail and finding the resulting vector that extends from the tail of the first vector to the head of the last vector.
Vector Addition Explained
Vector addition is not simply adding the magnitudes (sizes) of the vectors together. Instead, we consider both magnitude and direction. The reference material explains that the length of the arrow representing a vector indicates its size, and vectors of different sizes can be combined.
Graphical Method
- Head-to-Tail Placement: To add two or more vectors, place the tail of the second vector at the head of the first vector. If adding more than two vectors, continue placing each subsequent vector's tail at the head of the previous vector.
- Resultant Vector: Draw a new vector from the tail of the first vector to the head of the last vector. This new vector is the resultant vector or the sum of the vectors.
- Magnitude and Direction: The length of the resultant vector shows its magnitude, and the direction of the resultant vector shows the combined direction of the original vectors.
Example: Two Vectors
Let’s say you have two vectors:
- Vector A pointing towards the East with a magnitude of 3 units
- Vector B pointing towards the North with a magnitude of 4 units
- Draw vector A.
- Then at the head of vector A, draw vector B.
- Now draw a new vector going from the tail of vector A to the head of vector B. This resultant vector is the combination of A and B.
- The magnitude is calculated using Pythagorean theorem, which is 5 units, and it points towards the North-East.
Mathematical Method
Vector addition can also be done mathematically using component-wise addition if you know the vector components. This involves adding the corresponding x, y, and z components of the vectors to get the components of the resultant vector.
| Vector | X Component | Y Component |
|---|---|---|
| Vector A | Ax | Ay |
| Vector B | Bx | By |
| Resultant Vector | Ax+Bx | Ay+By |
Practical Insights
- Multiple Vectors: The graphical approach works for any number of vectors. Just keep placing them head-to-tail.
- Vector Size: As highlighted in the provided reference, the length of the vector arrow shows its size, allowing us to visually compare the relative magnitudes of different vectors.
- Applications: Vector addition is essential in physics, engineering, computer graphics, and many other fields. For instance, to determine the net force on an object, or the resultant displacement from multiple movements.
Example of Size in Combining Vectors
As the reference material from YouTube indicates, the length of the vector arrow indicates its size. Combining the vector 2 with vector 3, result in a vector that's longer than each individual vector. If combining the vector 5 with vector 2, the size of vector 5 is what affects the combined vector magnitude significantly.
