The component form of a vector

Definition

The component form of a vector expresses the vector in terms of its horizontal and vertical components. This representation is useful for performing vector addition, subtraction, and other operations.

5 Must Know Facts For Your Next Test

1. A vector in component form can be written as $\mathbf{v} = \langle v_x, v_y \rangle$, where $v_x$ and $v_y$ are the components.
2. The magnitude of a vector in component form can be found using $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$.
3. Vectors in component form can be added by summing their corresponding components: $\mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle$.
4. A unit vector in the same direction as a given vector $\mathbf{v}$ can be found by dividing each component by the magnitude: $\mathbf{\hat{v}} = \langle \frac{v_x}{|\mathbf{v}|}, \frac{v_y}{|\mathbf{v}|} \rangle$.
5. The dot product of two vectors in component form is calculated as $\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y$.

Review Questions

• What is the formula to find the magnitude of a vector given its components?
• How do you add two vectors when they are expressed in component form?
• What does the dot product of two vectors represent when calculated using their components?

Related terms

Magnitude: The length or size of a vector, often denoted as $|\mathbf{v}|$, calculated using the Pythagorean theorem for its components.

Dot Product: An algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number, representing their scalar product.

Unit Vector: A vector with a magnitude of one, used to indicate direction without regard to length.