Dot Product of Vectors.

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Presentation transcript:

Dot Product of Vectors

Quick Review

Quick Review Solutions

What you’ll learn about How to find the Dot Product How to find the Angle Between Vectors Projecting One Vector onto Another How to use vectors to find the work done by a force … and why Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.

Dot Product

Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. u·v=v·u u·u=||u||2 0·u=0 u·(v+w)=u·v+u·w (u+v)·w=u·w+v·w (cu) ·v=u·(cv)=c(u·v)

Example Finding the Dot Product

Example Finding the Dot Product

Example Finding the Dot Product

Example The dot product of u with itself is 5. What is the magnitude of u.

Angle Between Two Vectors This formula comes from the law of cosines!!

Example Finding the Angle Between Vectors

Example Finding the Angle Between Vectors

Example Finding the Angle Between Vectors

Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0. The terms orthogonal and perpendicular mean essentially the same thing – meeting at a right angle.

Example Are vectors u = <2,-3> and v = <6,4> orthogonal? Find the dot product. Therefore, yes. If you graph, you will see a right angle.

Projection of u and v

Work