Dot Product Second Type of Product Using Vectors

Dot Product Second Type of Product Using Vectors

Dot Product n 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 n v. w = a 1 a 2 + b 1 b 2 n The answer to a dot product is a number.

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

Angles Between Vectors n If u and v are two nonzero vectors, the angle θ, 0 ≤ θ ≤ p, between u and v is determine by the formula

Finding the Angle between Two Vectors n Example

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

Projection of a Vector onto Another Vector or Decomposition n 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. n 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.

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

Decomposition of v into v 1 and v 2 n The decomposition of v into v 1 and v 2, where v 1 is parallel to w and v 2 is perpendicular to w, is

Work Done by a Constant Force n n n 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.

Work Done by a Constant Force n Work done by a force using vectors is computed as