Dot Product

Second Type of Product Using Vectors

Dot Product

If v = a1i + b1j and w = a2i + b2j are two vectors, the dot product v . w is defined as

v . w = a1a2 + b1b2

The answer to a dot product is a number.

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

Angles Between Vectors

If u and v are two nonzero vectors, the angle θ, 0 ≤ θ ≤ , between u and v is determine by the formula

1cosu v

u v

Finding the Angle between Two Vectors Example

Navigation Problems

Finding the Actual Speed and Direction of an Aircraft

Example page 632

On-line Examples

Parallel and Orthogonal Vectors Two vectors are said to be parallel if the

angle between the two vectors is 0 or

Two vectors are orthogonal (at right angles), if the angle between the two nonzero vectors is /2 or the dot product is 0.

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.

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

1 2

v wv w

w

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Decomposition of v into v1 and v2 The decomposition of v into v1 and v2,

where v1 is parallel to w and v2 is perpendicular to w, is

1 2

v wv w

w

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2 1v v v ������������������������������������������

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.

Work Done by a Constant Force Work done by a force using vectors is

computed as

W F AB��������������