Dot Product

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

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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