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# Chapter 13 - PowerPoint PPT Presentation

Chapter 13. Section 13.3 The Dot Product. Dot Product The dot product of two vectors u and v is a number (a scalar) that can be computed in the following ways: Geometrically the dot product gives information about the angle  between the vectors u and v , specifically:. u. . v.

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Presentation Transcript

### Chapter 13

Section 13.3

The Dot Product

The dot product of two vectors u and v is a number (a scalar) that can be computed in the following ways:

Geometrically the dot product gives information about the angle  between the vectors u and v, specifically:

u

v

Orthogonal Vectors

If u and v are nonzero vectors the only way that is to have which means that (or ). This means the vectors are perpendicular which we call orthogonal. The vector u is orthogonal to vector v if and only if .

Algebraic Properties of Dot Product

Let u, v, and w be vectors and r a scalar.

and if and only if (Here  is the zero vector.)

If u and v are nonzero vectors if and only if

Find the angle between the vectors u and v given as:

and

Projections

The projection of a vector u onto a non zero vector v is a vector parallel to v whose difference with u is orthogonal to v.

To derive a formula for this let h be the length that v must be rescaled to get an orthogonal vector.

Multiply the unit vector in v’s direction by h to get the projection.

u

v

h

Example

Find all values for c so that the vectors u and v given to the right are perpendicular.

The idea is to find the dot product of u and v and set it equal to zero.

Setting equal to zero and solving:

We get the solutions and

Example

Show that for any two nonzero vectors u and v the two vectors v and are perpendicular vectors.

The formula for

Now take the dot product of v and and simplify it.