Advanced Precalculus Notes 8.5 The Dot Product

The dot product of two vectors is a scalar:

If v = 2i – 3j and w = 5i + 3j find:

a) v w∙ b) w v∙ c) v v∙ d) w w∙ e) ||v|| f) ||w||

2121 bbaa wv

Properties of dot products: Commutative Property: u v = v u∙ ∙ Distributive Property: u (v + w) = u v + u w∙ ∙ ∙ v v = ||v||∙ 2 0 v = 0∙ Angle between Vectors: Find the angle between u = 4i -3j and v = 2i + 5j

||||||||cos

vu

vu

A Boeing 737 aircraft maintains a constant airspeed of 500 mph due South. The velocity of the jet stream is 80 mph in a northeasterly direction. Find the actual speed and direction of the aircraft relative to the ground.

Parallel Vectors: if = 1 or -1, the vectors are parallel. Orthogonal Vectors: If the dot product is zero, the vectors are orthogonal (perpendicular). v = 3i – j w = 6i – 2j u = 2i – j z = 3i + 6j a) Are vectors v and w parallel, orthogonal of neither? b) Are vectors u and z parallel, orthogonal or neither?

|||||||| vu

vu

Vector projection of v onto w: Find the vector projection of v = i +3j onto w = i + j

ww

wvv

21 ||||

12 vvv

Work: work W done by a constant force F is:

W = (magnitude of force)(distance) =

An object is pulled with a force of 50 pounds. How much work is done in moving the object 100 feet if the handle makes an angle or 30º with the ground?

ABFABF

Assignment: page 626: 1 – 7, 12, 16, 17, 19, 21, 25, 27, 29, 32, 35