The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

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Presentation transcript:

The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

The dot product is useful for several things. One of the important uses is in a formula for finding the angle between two vectors that have the same initial point. u v  Technically there are two angles between these vectors, one going the "shortest" way and one going around the other way. We are talking about the smaller of the two.

Find the angle between the vectors v = 3i + 2j and w = 6i + 4j The vectors have the same direction. We say they are parallel because remember vectors can be moved around as long as you don't change magnitude or direction. What does it mean when the angle between the vectors is 0?

Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal. The vectors v and w are orthogonal. If the angle between 2 vectors is, what would their dot product be? Since cos is 0, the dot product must be 0. Vectors u and v in this case are called orthogonal. (similar to perpendicular but refers to vectors). compute their dot product and see if it is 0 w = 2i + 8j v = 4i - j

w v The projection of v onto w in the part of v in w's direction. Think of shining a light above v and the projection of v onto w would be the shadow formed.

What this does is takes a vector and breaks it up into two pieces---one in the direction of w and the other  /2 or 90° from it.

The work W done by a constant force F in moving an object from A to B is defined as Another use of the dot product is found in the formula below: This means the force is in some direction given by the vector F but the line of motion of the object is along a vector from A to B

Find the work done by a force of 50 pounds acting in the direction 3i + j in moving an object 20 feet from (0, 0) to (20, 0). 3i + j (20, 0)20i + 0j Let's find a unit vector in the direction 3i + j Remember to get a unit vector, divide a vector by it's magnitude Our force vector is in this direction but has a magnitude of 50 so we'll multiply our unit vector by 50.