Vectors in Component Form: Representation and Operations

Vectors in Component Form

Definition and Representation

Vectors are mathematical objects that have both magnitude and direction. In coordinate geometry, vectors are often represented in component form as ordered pairs or triples, depending on the dimension.

• Component Form: A vector from point A $(x_1, y_1)$ to point B $(x_2, y_2)$ is written as v = <$x_2 - x_1$, $y_2 - y_1$>.

• Graphical Representation: Vectors can be drawn as arrows on the coordinate plane, starting at the initial point and ending at the terminal point.

Finding the Component Form of a Vector

To find the component form of a vector given two points:

• Let $A(x_1, y_1)$ and $B(x_2, y_2)$ be two points.

• The vector from A to B is: $\vec{AB} = \langle x_2 - x_1, y_2 - y_1 \rangle$

• Example: If $A = (3, 5)$ and $B = (7, -4)$, then $\vec{AB} = \langle 7 - 3, -4 - 5 \rangle = \langle 4, -9 \rangle$

Operations with Vectors in Component Form

Vectors in component form can be added, subtracted, and multiplied by scalars:

• Addition: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$

• Subtraction: $\langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$

• Scalar Multiplication: $k \langle a, b \rangle = \langle ka, kb \rangle$

Example Calculation

Given $\vec{v} = \langle 5, 7 \rangle$ and $\vec{w} = \langle 4, 7 \rangle$:

• Subtraction: $\vec{v} - \vec{w} = \langle 5, 7 \rangle - \langle 4, 7 \rangle = \langle 5 - 4, 7 - 7 \rangle = \langle 1, 0 \rangle$

Applications

• Vectors are used in physics to represent quantities such as force and velocity.

• In trigonometry, vectors are foundational for understanding vector addition, direction, and magnitude.

Summary Table: Vector Operations

Operation	Formula	Example
Addition	$\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$	$\langle 2, 3 \rangle + \langle 1, 4 \rangle = \langle 3, 7 \rangle$
Subtraction	$\langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$	$\langle 5, 7 \rangle - \langle 4, 7 \rangle = \langle 1, 0 \rangle$
Scalar Multiplication	$k \langle a, b \rangle = \langle ka, kb \rangle$	$2 \langle 3, -1 \rangle = \langle 6, -2 \rangle$

Additional info: Vectors in component form are essential for later topics in trigonometry, such as dot product, projections, and applications in physics and engineering.