To convert a vector from its initial point to its terminal point into component form, you find the difference between the corresponding coordinates of the terminal and initial points.
When a vector is described as starting at point $P(x_1, y_1)$ and ending at point $Q(x_2, y_2)$, its component form is $\langle x_2 - x_1, y_2 - y_1 \rangle$. This standard method is used to represent a vector uniquely regardless of its starting position, as long as its magnitude and direction are the same.
Understanding Component Form
The component form of a 2D vector is typically written as $\langle a, b \rangle$, where 'a' represents the horizontal change (movement along the x-axis) and 'b' represents the vertical change (movement along the y-axis) from the vector's tail to its head. For a 3D vector, it would be $\langle a, b, c \rangle$.
Steps to Convert Using Initial and Terminal Points
As suggested by sources like the YouTube reference snippet mentioning the need for $x_1, y_1, x_2,$ and $y_2$ coordinates, the most common conversion involves knowing the vector's start and end points.
Here are the steps:
- Identify the Coordinates: Determine the coordinates of the initial point ($P$) and the terminal point ($Q$) of the vector. Let $P = (x_1, y_1)$ and $Q = (x_2, y_2)$.
- Calculate the Horizontal Component: Subtract the x-coordinate of the initial point ($x_1$) from the x-coordinate of the terminal point ($x_2$). This gives you the horizontal component, $a = x_2 - x_1$.
- Calculate the Vertical Component: Subtract the y-coordinate of the initial point ($y_1$) from the y-coordinate of the terminal point ($y_2$). This gives you the vertical component, $b = y_2 - y_1$.
- Write in Component Form: Combine the horizontal and vertical components into the component form: $\langle a, b \rangle = \langle x_2 - x_1, y_2 - y_1 \rangle$.
If working in three dimensions with points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, the component form is $\langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$.
Example
Let's convert a vector that starts at point $P(3, 2)$ and ends at point $Q(7, 5)$ into component form.
- Initial Point $P$: $(x_1, y_1) = (3, 2)$
- Terminal Point $Q$: $(x_2, y_2) = (7, 5)$
Using the steps:
- $x_1 = 3$, $y_1 = 2$, $x_2 = 7$, $y_2 = 5$.
- Horizontal Component: $a = x_2 - x_1 = 7 - 3 = 4$.
- Vertical Component: $b = y_2 - y_1 = 5 - 2 = 3$.
- Component Form: $\langle a, b \rangle = \langle 4, 3 \rangle$.
So, the vector from $P(3, 2)$ to $Q(7, 5)$ in component form is $\langle 4, 3 \rangle$.
Example Data
| Point | Coordinates |
|---|---|
| Initial Point P | $(x_1, y_1)$ |
| Terminal Point Q | $(x_2, y_2)$ |
The component form is $\langle x_2 - x_1, y_2 - y_1 \rangle$.
Converting a vector from its initial and terminal points to component form is a fundamental operation in vector algebra, providing a standardized way to represent vector displacement.
