The component form of the resultant vector, when adding two vectors $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}$, is given by adding their corresponding components. The resultant vector, denoted as $\mathbf{a} + \mathbf{b}$, has the component form:
$(a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j}$
Alternatively, this can be written in angle bracket notation as $<a_1 + b_1, a_2 + b_2>$.
Understanding Resultant Vectors
Based on the reference provided, when a vector changes direction from vector a to vector b, the resultant vector represents the combined effect or the overall displacement. The reference explicitly states that the resultant vector is the sum of the two vectors a+b.
Vectors in component form break down the vector's magnitude and direction into its horizontal and vertical movements. For a 2D vector, these are typically represented along the x-axis (using the unit vector i) and the y-axis (using the unit vector j).
If vector a is $a_1\mathbf{i} + a_2\mathbf{j}$, it means a moves $a_1$ units horizontally and $a_2$ units vertically. Similarly, for vector b = $b_1\mathbf{i} + b_2\mathbf{j}$, it moves $b_1$ units horizontally and $b_2$ units vertically.
How to Add Vectors in Component Form
Adding vectors in component form is a straightforward process. To find the resultant vector r = a + b, you simply add the corresponding components of the individual vectors.
- Add the horizontal components: Sum the coefficients of the i terms ($a_1$ and $b_1$). This gives the horizontal component of the resultant vector: $(a_1 + b_1)\mathbf{i}$.
- Add the vertical components: Sum the coefficients of the j terms ($a_2$ and $b_2$). This gives the vertical component of the resultant vector: $(a_2 + b_2)\mathbf{j}$.
Combining these two results gives you the component form of the resultant vector:
$\mathbf{r} = (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j}$
This method directly incorporates the principle from the reference that the resultant is the sum a+b by performing that sum on the individual components.
Example of Finding the Resultant Vector
Let's consider an example to illustrate this concept.
Suppose vector a = $3\mathbf{i} + 4\mathbf{j}$ and vector b = $2\mathbf{i} - 1\mathbf{j}$.
According to the principle of adding vectors in component form:
- Horizontal component of resultant: $3 + 2 = 5$
- Vertical component of resultant: $4 + (-1) = 3$
So, the resultant vector is $5\mathbf{i} + 3\mathbf{j}$.
Here's how this looks in a simple table:
| Vector | i component ($a_1$ or $b_1$) | j component ($a_2$ or $b_2$) |
|---|---|---|
| a | 3 | 4 |
| b | 2 | -1 |
| a+b | $3 + 2 = 5$ | $4 + (-1) = 3$ |
The resultant vector is $5\mathbf{i} + 3\mathbf{j}$ or $<5, 3>$.
This simple method extends to vectors in three dimensions ($a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$) by adding the corresponding k components as well. The core principle remains the same: add the respective components.
