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Dot Product

Dot Product. Second Type of Product Using Vectors. Dot Product. If v = a 1 i + b 1 j and w = a 2 i + b 2 j are two vectors, the dot product v . w is defined as v . w = a 1 a 2 + b 1 b 2 The answer to a dot product is a number. Properties of the Dot Product.

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Dot Product

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  1. Dot Product Second Type of Product Using Vectors

  2. Dot Product • If v = a1i + b1j and w = a2i + b2j are two vectors, the dot productv . w is defined as • v . w = a1a2 + b1b2 • The answer to a dot product is a number.

  3. Properties of the Dot Product • If u, v, and w are vectors, then • Commutative Property • u . v = v . u • Distributive Property • u . (v + w) = u . v + u . w • v . v = ||v||2 • 0 . v = 0

  4. Angles Between Vectors • If u and v are two nonzero vectors, the angle θ, 0 ≤ θ ≤ p, between u and vis determine by the formula

  5. Finding the Angle between Two Vectors • Example

  6. Navigation Problems Finding the Actual Speed and Direction of an Aircraft Example page 632 On-line Examples

  7. Parallel and Orthogonal Vectors • Two vectors are said to be parallel if the angle between the two vectors is 0 or p • Two vectors are orthogonal (at right angles), if the angle between the two nonzero vectors is p/2 or the dot product is 0.

  8. Projection of a Vector onto Another Vector or Decomposition • Vector Projection allows us to find “how much” of the magnitude is working in the horizontal direction and “how much” is working in the vertical direction. • We decompose the one vector into a vector that is parallel to the vector we are projecting onto and one that is orthogonal to the vector we are projecting onto.

  9. Vector Projection • Remember that we will always have two vectors when we are through. • If v and w are two nonzero vectors, the vector projection of v onto w is

  10. Decomposition of v into v1and v2 • The decomposition of v into v1and v2, where v1 is parallel to w and v2 is perpendicular to w, is

  11. Work Done by a Constant Force • Work = (magnitude of force) (distance) • Up till now all work you have been computing has been at an angle of 90 degrees or 0 degrees. • Vectors allow us to push or pull at any angle.

  12. Work Done by a Constant Force • Work done by a force using vectors is computed as

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