25. Dot Product of Vectors

Dot Product

Example

Example

Example

Applications of dot product Finding the angle between vectors Determining if vectors are orthogonal (perpendicular) or parallel Projecting a vector onto another Work

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

Example

Example

Orthogonal Vectors Parallel Vectors The vectors u and v are orthogonal if u·v = 0. Parallel Vectors The vectors u and v are parallel if u·v =

Example Are vectors u = <2,-3> and v = <6,4> orthogonal, parallel, or neither? Orthogonal

Example – Tell if vectors are orthogonal, parallel, or neither

Projecting a vector onto another We have seen applications of finding a resultant vector such as forces pulling on an object or wind resistance on a plane There are other applications in physics and engineering where you need to do the reverse – decompose the vector into the sum of 2 perpendicular vector components

Vector components Consider a boat on an inclined ramp shown below. The force F due to gravity pulls the boat down the ramp (w1) and against the ramp (w2) . Notice that w1 and w2 are orthogonal. These are called vector components.

To find w1 and w2 (the vector components) w1 is the projection of u onto v and is denoted w1=projvu w2 = u - w1 The projection (w1) is like shining a light onto a vector and finding its shadow on the second vector

Example Find the projection u = <3,-5> onto v = <6,2>. Then write u as the sum of two vector components.

Work Work is the product of the force acting in the direction something moves and the distance it travels There are 2 ways to find this – we will only look at one – the dot product Find the component form of the force and the component form of the displacement and then find their dot product W =

Example Find the work done by a force F = 4i + 9j in moving an object from (4,6) to (8, 7)

Example Find the work done by a force of 53 N at 47o in moving an object 36 m horizontally.