Section 10.7. Multiply like components together, then add u 1 v 1 + u 2 v 2 Length of a vector:…
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Section 10.7
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Two nonzero vectors u and v are orthogonal iff their dot product is zero.
Transcript of Section 10.7. Multiply like components together, then add u 1 v 1 + u 2 v 2 Length of a vector:…
Section 10.7
Multiply like components together, then addu1v1 + u2v2
Length of a vector:
Angle between Vectors Theorem:
uuu ||||
)|||| ||||
(cos 1
vuvu
Two nonzero vectors u and v are orthogonal iff their dot product is zero.
A passenger jet is cruising at a speed of 300 knots in a direction 25˚ east of south and there is a 40 knot wind coming from the southwest. By how many degrees will the jet be thrown off course?
Jet’s velocity vector: [300, 295] Wind velocity vector: [40, 45]
Component: 300cos295 + 40 cos 45 300 sin 295 + 40 sin 45
Component: 300cos295 + 40 cos 45 300 sin 295 + 40 sin 45
(155.0697, -243.608)
Put back in polar form:
R = 288.776θ = -57.5˚
So the plane is getting “pushed” 7.48˚ off course.
Calculate the dot product of <-1, 8> and <-5, -10.2>
-1 * -5 + 8* -10.25 – 81.6
-76.6
Find the measure of the angle between the vectors u = <-9, 12> and v = <-4, 3>
Formula: )|||| ||||
(cos 1
vuvu
)5*15
72(cos 1 3.16
Explain the relationship between the dot product of two vectors and the angle between them.
If the dot product is positive, the angle is acute
If the dot product is negative, the angle is obtuse
If the dot product is zero, the angle is right.
Describe all vectors in the plane that are orthogonal to <17,5> and have length 4.
Two equations: 0 = 17v1 + 5v2
16 = v12 + v2
2
Solve for v2 in the first: v2 = -17/5 v1
Substitute back in: 16 = v12 + (-17/5v1)2
16 = v12 + (289/25)v1
2 16 = 314/25v1
2
400/314 = v12
131420
v 131420
v
Two equations: 0 = 17v1 + 5v2
16 = v12 + v2
2
Solve for v2 in the first: v2 = -17/5 v1
Plug back in:
231420
517
v
231468
v
231420
517
v
231468
v
Page 6251 – 9, 11,
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