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Vector

from class:

Algebra and Trigonometry

Definition

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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5 Must Know Facts For Your Next Test

  1. Vectors can be added using the head-to-tail method or by component-wise addition.
  2. The magnitude of a vector $\mathbf{v} = \langle x, y \rangle$ is calculated as $|\mathbf{v}| = \sqrt{x^2 + y^2}$.
  3. The direction of a vector is given by the angle it makes with the positive x-axis, often found using $\theta = \tan^{-1}(y/x)$.
  4. 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 $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2$.
  5. 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?
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