6.2 Dot Product of Vectors

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

Example Finding the Dot Product

Example Finding the 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 5. (cu) ·v = u·(cv) = c(u·v)

Angle Between Two Vectors

Find the angle between the vectors u = <2, 3> and v = <-2, 5>

You Try!

Example Finding the Angle Between Vectors

Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0. The terms “perpendicular” and “orthogonal” almost mean the same thing. The zero vector has no direction angle, so technically speaking, the zero vector is not perpendicular to any vector However, the zero vector is orthogonal to every vector. Except for this special case, orthogonal and perpendicular are the same

Prove that the vectors u = <2, 3> and v = <-6, 4> are orthogonal

Projection of u and v

Find the vector projection of u = <5, 2> onto v = <10, 0>.

Find the vector projection of u = <6, 2> onto v = <5, -5> Find the vector projection of u = <6, 2> onto v = <5, -5>. Then write u as the sum of two orthogonal vectors, one of which is projvu.

Work

Find the work done by a 10 pound force acting in the direction <1, 2> in moving an object 3 feet from (0, 0) to (3, 0).