A vector is a quantity that has both magnitude and direction. It can be represented graphically as an arrow or algebraically as an ordered pair or triplet.
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Vectors can be added using the head-to-tail method or by component-wise addition.
The magnitude of a vector $\mathbf{v} = \langle x, y \rangle$ is calculated as $|\mathbf{v}| = \sqrt{x^2 + y^2}$.
The direction of a vector is given by the angle it makes with the positive x-axis, often found using $\theta = \tan^{-1}(y/x)$.
Dot product of two vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$ is computed as $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2$.
Cross product applies to three-dimensional vectors and results in another vector perpendicular to both original vectors.
Review Questions
How do you calculate the magnitude of a vector given its components?
What method can be used to add two vectors graphically?
What does the dot product of two vectors represent?
Related terms
Magnitude: The length or size of a vector, calculated using the Pythagorean theorem for its components.
Dot Product: An operation that takes two equal-length sequences of numbers and returns a single number. It is calculated as the sum of the products of corresponding entries.
Cross Product: A binary operation on two vectors in three-dimensional space resulting in another vector perpendicular to both.