Unit vectors of the axes

Definition

Unit vectors of the axes are vectors that have a magnitude of 1 and point in the direction of the coordinate axes. They are typically denoted as $\hat{i}$, $\hat{j}$, and $\hat{k}$ in three-dimensional space for the x, y, and z-axes respectively.

5 Must Know Facts For Your Next Test

1. Unit vectors are dimensionless and have a magnitude of exactly 1.
2. $\hat{i}$ is the unit vector along the x-axis, $\hat{j}$ is along the y-axis, and $\hat{k}$ is along the z-axis.
3. Any vector in three-dimensional space can be expressed as a linear combination of $\hat{i}$, $\hat{j}$, and $\hat{k}$.
4. The dot product between any two different unit vectors (e.g., $\hat{i} \cdot \hat{j}$) is zero because they are orthogonal.
5. The cross product between any two different unit vectors (e.g., $\hat{i} \times \hat{j} = \hat{k}$) results in another unit vector.

Review Questions

• What is the magnitude of a unit vector?
• How do you represent a vector in three-dimensional space using unit vectors?
• What is the result of the dot product between two perpendicular unit vectors?

Related terms

Magnitude: The length or size of a vector; it is calculated as the square root of the sum of its components squared.

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

Cross Product: A binary operation on two vectors in three-dimensional space that results in another vector which is perpendicular to both input vectors.