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The Dot Product

The Dot Product. Definition of a Dot Product. If v=a 1 i+b 1 j and w = a 2 i+b 2 j are vectors, the dot product is defined as. The dot product of two vectors is the sum of the products of their horizontal and vertical components. If v = 5 i – 2 j and w = -3 i + 4 j , find:

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The Dot Product

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  1. The Dot Product

  2. Definition of a Dot Product If v=a1i+b1j and w = a2i+b2j are vectors, the dot product is defined as The dot product of two vectors is the sum of the products of their horizontal and vertical components.

  3. If v = 5i – 2j and w = -3i + 4j, find: a. v · wb. w · vc. v · v. Text Example Solution To find each dot product, multiply the two horizontal components, and then multiply the two vertical components. Finally, add the two products. a. v · w = 5(-3) + (-2)(4) = -15 – 8 = -23 b. w · v = (-3)(5) + (4)(-2) = -15 – 8 = -23 c. v · v = (5)(5) + (-2)(-2) = 25 + 4 = 29

  4. If u, v, and w, are vectors, and c is a scalar, then 1. u · v = v · u 2. u · (v + w)= u · v + u · w 3. 0 · v = 0 4. v · v = || v ||2 5.(cu) · v = c(u · v) = u · (cv) Properties of the Dot Product

  5. Alternative Formula for the Dot Product • If v and w are two nonzero vectors and  is the smallest nonnegative angle between them, then • v · w = ||v|| ||w|| cos.

  6. Alternative Formula for the Dot Product • If v and w are two nonzero vectors and  is the smallest nonnegative angle between them, then

  7. Formula for the Angle between Two Vectors • If v and w are two nonzero vectors and  is the smallest nonnegative angle between v and w, then

  8. Example • Find the angle  between v=2i-4j and w=3i+2j. Solution:

  9. The Dot Product and Orthogonal Vectors • Two nonzero vectors v and w are orthogonal if and only if v•w=o. Because 0•v=0, the zero vector is orthogonal to every vector v.

  10. Example • Are the vectors v=3i-2j and w=3i+2j orthogonal? Solution: The vectors are not orthogonal.

  11. The Vector Projection of v Onto w • If v and w are two nonzero vectors, the vector projection of v onto w is

  12. Example • If v=3i+4j and w=2i-5j, find the projection of v onto w Solution:

  13. The Vector Components of v • Let v and w be two nonzero vectors. Vector v can be expressed as the sum of two orthogonal vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. Thus, v = v1 + v2. The vectors v1 and v2 are called the vector components of v. The process of expressing v as v1 and v2 is called the decomposition of v into v1 and v2.

  14. Example • Let v=3i+j and w=2i-3j. Decompose v into two vectors, where one is parallel to w and the other is orthogonal to w. Solution:

  15. Definition of Work • The work W done by a force F in moving an object from A to B is • W = F · AB.

  16. The Dot Product

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