6.3 Vectors.notebook
March 19, 2012
6.3 Vectors
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6.3 Vectors.notebook
March 19, 2012
Vector: has magnitude (length) and direction
Q (x2, y2)
PQ
terminal point
||v||
magnitude
P (x1, y1)
initial point
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6.3 Vectors.notebook
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Component Form of a Vector A vector whose initial point is at the origin (0, 0) can be represented by the coordinates of its terminal point (v1, v2). v = < v1 , v2 >
The component form of the vector with initial point P = (x1, y1) and terminal point Q = (x2, y2) is:
The magnitude (or length) of v is:
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6.3 Vectors.notebook
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Example 1: Find the component form and magnitude of the vector v that has (1, 7) as its initial point and (4, 3) as its terminal point.
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6.3 Vectors.notebook
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Vector Operations Geometrically, the product of a vector v and a scalar k is . . . the vector that is k times as long as v
If k is positive, kv has the same direction as v, and if k is negative, kv has the opposite direction.
v
1/2v
2v
­v
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6.3 Vectors.notebook
To add two vectors geometrically, position them so the initial point of one coincides with the terminal point of the other
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v
u
This technique is called the parallelogram law for vector addition because the vector u + v, often called the resultant of vector addition, is . . . the diagonal of a parallelogram with u and v as sides 6
6.3 Vectors.notebook
March 19, 2012
Let u = 〈u1, u2〉 and v = 〈v1, v2〉 be vectors and let k be a scalar (vector addition)
ku = < ku1 , ku2 > (Scalar multiplication)
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6.3 Vectors.notebook
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Let u = 〈1, 6〉 and v = 〈− 4, 2〉. Sketch the operations geometrically. Then find:
(a) 3v
(b) u + v
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6.3 Vectors.notebook
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Unit Vector: has a magnitude of 1 To find a unit vector u that has the same direction as vector v . . . u = v ||v||
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6.3 Vectors.notebook
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Find a unit vector in the direction of v = 〈− 8, 6〉.
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6.3 Vectors.notebook
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Find the vector v with the given magnitude and same direction as u.
||v|| = 3, u = < 4, -4 >
find the unit vector
multiply it by the magnitude
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6.3 Vectors.notebook
Standard Unit Vectors:
i = <1, 0> j = <0, 1>
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Linear Combination
Vector v = 〈v1, v2〉
can also be represented as: j = <0, 1>
v = v1i + v2j i = <1, 0> 12
6.3 Vectors.notebook
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Let v = 〈− 5, 3〉. Write v as a linear combination of the standard unit vectors i and j.
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6.3 Vectors.notebook
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The initial point of a vector is (-6, 4) and the terminal point is (0, 1)
Write a linear combination of the standard vector.
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6.3 Vectors.notebook
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Find the component form of v and sketch the specified vector
operations geometrically.
w = i + 2j
u = 2i - j
v = -u + 2w
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6.3 Vectors.notebook
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(x, y)
Direction Angles
u
Direction Angle: measured counterclockwise from the x axis to terminal point of u
y = sinθ
θ
x = cosθ
u = 〈x, y〉 = < cosθ , sinθ >
= (cosθ)i + (sinθ)j
v = ||v|| < cosθ , sinθ > = ||v|| cosθ i + ||v|| sinθ j Since v = ai + bj, tan θ = b (like a reference angle)
a
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6.3 Vectors.notebook
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Find the component form of V given its magnitude and angle. ||v|| = 4, θ = 45o
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6.3 Vectors.notebook
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Let v = − 4i + 5j. Find the magnitude and direction angle for v.
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6.3 Vectors.notebook
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Applications of Vectors
force / velocity / tension = magnitude / length
also think: Geometry, right triangle trig, law of cosines and sines
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6.3 Vectors.notebook
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Using Vectors to Find Speed and Direction Example 10 p.416
An airplane is traveling at a speed of 500 mph with a bearing of 330o. The airplane encounters a wind blowing 70 mph in the direction N 45o E. What are the resultant speed and direction of the airplane? 20
6.3 Vectors.notebook
March 19, 2012
Highlight of Important Vocab / Formulas:
Component Form: v = < v1, v2 > found by subtracting x values and y values of points
Magnitude: ||v||
found by: √(v1 + v22) or the distance formula
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Unit Vector: v ||v||
Linear Combination of v: v1i + v2j Direction Angles: v = ||v|| cosθ i + ||v|| sinθ j
If v = ai + bj, then tan θ = b
a
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