Vector components

Definition

Vector components are the projections of a vector along the axes of a coordinate system. They simplify vector calculations by breaking vectors into perpendicular directions.

5 Must Know Facts For Your Next Test

1. The components of a vector in two dimensions are found using trigonometric functions: $V_x = V \cos(\theta)$ and $V_y = V \sin(\theta)$.
2. In three dimensions, a vector can be broken down into x, y, and z components.
3. Vector addition can be performed by separately adding the corresponding components of each vector.
4. The magnitude of a vector can be found using its components: $|\mathbf{V}| = \sqrt{V_x^2 + V_y^2}$ in two dimensions and $|\mathbf{V}| = \sqrt{V_x^2 + V_y^2 + V_z^2}$ in three dimensions.
5. Unit vectors along the axes (i.e., i, j, k) are used to express vector components in Cartesian coordinates.

Review Questions

• What trigonometric functions are used to find the x and y components of a vector?
• How do you calculate the magnitude of a vector given its components?
• What is the significance of unit vectors i, j, and k in expressing vector components?

Related terms

Magnitude:

The length or size of a vector; calculated as $|\mathbf{V}| = \sqrt{V_x^2 + V_y^2}$ for 2D or $|\mathbf{V}| = \sqrt{V_x^2 + V_y^2 + V_z^2}$ for 3D.

Unit Vector:

A vector with a magnitude of 1 that indicates direction only; denoted as i, j, k along the x, y, z axes respectively.

Scalar Projection: The length (magnitude) of the projection of one vector onto another; calculated using dot product divided by the magnitude of the other vector.