6.2 Dot products comp.notebook 1 February 04, 2020 Feb 157:57 PM 6.2 Dot Product of Vectors Dot product or inner product u• v =u 1 v 1 +u 2 v 2 Properties of Dot Products Let u,v and w be vectors and c a scalar. 1) u• v = v• u 4) u• (v+w)=u• v+u• w 2) u• u =|u| 2 (u+v)• w =u• w+v• w 3) 0• u=0 5) (cu)• v =u• (cv)=c(u• v)

6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

Download PDF Report

Upload
others
View
10
Download
0

Embed Size (px)

Citation preview

Page 1: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Feb 157:57 PM

6.2 Dot Product of Vectors

Dot product or inner productu•v = u1v1+u2v2

Properties of Dot Products

Let u,v and w be vectors and c a scalar.1) u•v = v•u 4) u•(v+w)=u•v+u•w2) u•u = |u|2 (u+v)•w =u•w+v•w3) 0•u=0 5) (cu)•v =u•(cv)=c(u•v)

Page 2: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Feb 157:57 PM

Ex. 1Find the dot product of (a) <3,1>•<2,5> (b) (2i +2 j) • (3i – 5j)

Ex. 2Find the dot product to find the length of the vector u=<5,2>.

Page 3: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Feb 157:57 PM

Angle Between Two vectors

Let u and v be two nonzero vectors in standard position. The angle θ, between u and v lies between 0 and 180o.

cosθ= and so θ=

Ex. 3Find the angle between the vectors u and v if a) u=<2,3> and v=<4,1>

Page 4: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Feb 157:57 PM

Perpendicular vectorsIf u and v are perpendicular, then the angle between

them is 90o and

Orthogonal vectorsVectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably with the exception – when we talk about the zero vector.

Parallel vectorsVectors u and v are parallel if they are multiples of one another.

Ex.4 Given u=<6,3> and v=<5,10>, determine if u and v parallel, perpendicular or neither.

TRYGiven u=<2,3> and v=<4,6>, determine if u and v parallel, perpendicular or neither.

Page 5: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Feb 157:57 PM

Ex. 5 Kate is sitting on a sled on the side of a hill inclined at 55o. The combined weight of Kate and the sled is 140lb. What force is required for Jack to keep the sled from sliding down the hill? What is the force perpendicular to the hill?

Page 6: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Feb 157:57 PM

Work Done

Work done is directly proportional to the constant force and the distance moved.

Ex.6 Find the work done lifting a 50lb chair 5ft straight up.

Ex. 7The angle between a 35lb force F and CD is 320 C=(4,5) to D=(8,11). Find the work done in moving the force from C to D.

Ex.8Suppose Tanvin is playing tug of war with Kano his dog. If Tanvin exerts a force of 52lb in a direction of 47o above the horizontal and Kano exerts a force of 37lb in a direction of 22o below the horizontal, what is the resultant force?

Page 7: 6.2 Dot Product of Vectors - Amphitheater Public Schools · Orthogonal vectors Vectors u and v are orthogonal iff u•v=0. We can use orthogonal and perpendicular interchangeably

6.2 Dot products comp.notebook

February 04, 2020

Mar 88:45 PM

Ex.9 Find the work done by a 10lb force acting in the direction <1,2> in moving an object 3 feet from (0,0) to (3,0).

Ex. 10Find the work done by a force F of 40lbs acting in the direction <8,6> in moving an object 5ft from (0,0) to a point in the first quadrant along the line y=3x.