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A vector is a quantity with two independent properties: magnitude and direction A vector can also be defined as a directed line segment having an originating point (beginning) and a terminating point (end )
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What s the standard form of a vector in the xy plane What s the standard form of a vector in the polar plane What s the advantage of putting a vector into its standard form
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In any coordinate system, a vector is in standard form if and only if its originating point is at the coordinate origin The standard form allows us to uniquely define a vector as an ordered pair that represents the coordinates of its terminating point alone
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How can we find the magnitude of a standard-form vector b in the xy plane whose terminating point has the coordinates (xb,yb)
Answer 4-3
The magnitude of b, which we can write as rb, is found by using the formula for the distance of the terminating point from the origin In this case, we get rb = (xb2 + yb2)1/2 In some texts, the magnitude of b would be denoted as |b| or b
Question 4-4
How can we find the direction of a standard-form vector b in the xy plane whose terminating point has the coordinates (xb,yb)
Part One Answer 4-4
We find the polar direction angle of the point (xb,yb) If we call this angle qb, the process can be broken down into the following nine possible cases: If xb = 0 and yb = 0, then qb = 0 by default If xb > 0 and yb = 0, then qb = 0 If xb > 0 and yb > 0, then qb = Arctan ( yb/xb) If xb = 0 and yb > 0, then qb = p /2 If xb < 0 and yb > 0, then qb = p + Arctan ( yb/xb) If xb < 0 and yb = 0, then qb = p If xb < 0 and yb < 0, then qb = p + Arctan ( yb/xb) If xb = 0 and yb < 0, then qb = 3p /2 If xb > 0 and yb < 0, then qb = 2p + Arctan ( yb/xb)
In some texts, the direction of b is denoted as dir b
Question 4-5
Imagine two vectors a and b in the xy plane, in standard form with terminating-point coordinates a = (xa,ya) and b = (xb,yb) How can we find the sum of these vectors
Answer 4-5
We calculate the sum vector a + b using the formula a + b = [(xa + xb),( ya + yb)]
Question 4-6
How can we calculate the Cartesian negative of a vector that s in standard form How does the Cartesian negative compare with the original vector
Answer 4-6
We take the negatives of both coordinate values For example, if we have
b = (xb,yb)
then its Cartesian negative is -b = ( xb, yb) The Cartesian negative has the same magnitude as the original vector, but points in the opposite direction
Review Questions and Answers Question 4-7
Imagine two Cartesian vectors a and b, in standard form with terminating-point coordinates a = (xa,ya) and b = (xb,yb) How can we find a b How can we find b a How do these two vectors compare
Answer 4-7
We calculate the difference vector a b using the formula a b = [(xa xb),( ya yb)] We find difference vector b a by reversing the order of subtraction for each coordinate, getting b a = [(xb xa),( yb ya)] In the Cartesian plane, the difference vector b a is always equal to the negative of the difference vector a b
Question 4-8
Suppose we have a vector expressed in polar form as c = (qc,rc) where qc is the direction angle of c, and rc is the magnitude of c How can we convert c to a standard-form vector (xc,yc) in the Cartesian plane
Answer 4-8
We use formulas adapted from the polar-to-Cartesian conversion We get (xc,yc) = [(rc cos qc),(rc sin qc)]
Question 4-9
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