College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
Vectors are quantities that have both magnitude and direction, often represented graphically by arrows. Common examples include displacement, velocity, and force.
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Vectors can be added using the head-to-tail method or by breaking them into components.
The magnitude of a vector can be found using the Pythagorean theorem for its components: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$.
Unit vectors are vectors with a magnitude of one and indicate direction only.
Vector subtraction can be performed by adding the negative of the vector to be subtracted.
The dot product of two vectors is a scalar given by $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)$, where $\theta$ is the angle between the vectors.
Review Questions
What is the difference between scalar and vector quantities?
How do you find the resultant vector when adding two vectors graphically?
Explain how to calculate the dot product of two vectors.
Related terms
Scalar: A quantity that has only magnitude and no direction, such as temperature or mass.
Unit Vector: A vector with a magnitude of one, used to specify direction.
Dot Product: $\vec{A} \cdot \vec{B}$ results in a scalar calculated as $|\vec{A}| |\vec{B}| \cos(\theta)$, quantifying how much one vector extends in the direction of another.