Angle between two vectors

Definition

The angle between two vectors is the measure of the smallest rotation required to align one vector with the other. It can be calculated using the dot product and magnitudes of the vectors.

5 Must Know Facts For Your Next Test

1. The angle between two vectors $\vec{A}$ and $\vec{B}$ can be found using the formula: $\cos(\theta) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$.
2. The value of $\cos(\theta)$ ranges from -1 to 1, where $\theta$ is between 0 and 180 degrees.
3. When two vectors are perpendicular (orthogonal), their dot product is zero, meaning $\cos(90^\circ) = 0$.
4. If two vectors are parallel, their angle is either 0 degrees (same direction) or 180 degrees (opposite direction).
5. The dot product is commutative; hence, the formula for calculating the angle between two vectors remains consistent regardless of their order.

Review Questions

• What formula is used to calculate the angle between two vectors?
• What does a dot product of zero signify about the angle between two vectors?
• What are the possible angles when two vectors are parallel?

Related terms

Dot Product:

An algebraic operation that takes two equal-length sequences of numbers and returns a single number, given by $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)$.

Magnitude:

The length or size of a vector, denoted as $|\vec{A}|$, calculated as $|\vec{A}| = \sqrt{x^2 + y^2 + z^2}$ for a vector in three-dimensional space.

Orthogonal Vectors: Two vectors that are perpendicular to each other. Their dot product equals zero.