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We ve seen how vectors add and subtract in two dimensions In this chapter, we ll learn how to multiply a vector by a real number Then we ll explore two different ways in which vectors can be multiplied by each other
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The simplest form of vector multiplication involves changing the magnitude by a real-number factor called a scalar A scalar is a one-dimensional quantity that can be positive, negative, or zero If the scalar is positive, the vector direction stays the same If the scalar is negative, the vector direction reverses If the scalar is zero, the vector disappears
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Cartesian vector times positive scalar Imagine a standard-form vector a in the Cartesian xy plane, defined by an ordered pair whose coordinates are xa and ya, so that
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a = (xa,ya) Suppose that we multiply a positive scalar k+ by each of the vector coordinates individually, getting two new coordinates Mathematically, we write this as k+a = (k+ xa,k+ ya) This vector is called the left-hand Cartesian product of k+ and a If we multiply both original coordinates on the right by k+ instead, we get a k+ = (xak+,yak+)
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That s the right-hand Cartesian product of a and k+ The individual coordinates of k+a and ak+ are products of real numbers We learned in pre-algebra that real-number multiplication is commutative, so it follows that k+a = (k+ xa,k+ya) = (xak+,yak+) = a k+ We ve just shown that multiplication of a Cartesian-plane vector by a positive scalar is commutative We don t have to worry about whether we multiply on the left or the right; we can simply talk about the Cartesian product of the vector and the positive scalar
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An example Figure 5-1 illustrates the Cartesian vector ( 1, 2) as a solid, arrowed line segment If we multiply this vector by 3 on the left, we get
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3 ( 1, 2) = {[3 ( 1)],[3 ( 2)]} = ( 3, 6) If we multiply the original vector by 3 on the right, we get ( 1, 2) 3 = [( 1 3)],( 2 3)] = ( 3, 6) The new vector is shown as a dashed, gray, arrowed line segment pointing in the same direction as the original vector, but 3 times as long
Figure 5-1 Cartesian products of the scalars 3 and 3
with the vector ( 1, 2)
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Cartesian vector times negative scalar Now suppose we want to multiply a by a negative scalar instead of a positive scalar Let s call the scalar k The left-hand Cartesian product of k and a is
k a = (k xa,k ya) The right-hand Cartesian product is a k = (xak ,yak ) As with the positive constant, the commutative property of real-number multiplication tells us that k a = (k xa,k ya) = (xak ,yak ) = ak We don t have to worry about whether we multiply on the left or the right We get the same result either way
An example Once again, look at Fig 5-1 with the vector ( 1, 2) shown as a solid, arrowed line segment When we multiply it by the scalar 3 on the left, we obtain
3 ( 1, 2) = {[ 3 ( 1)],[ 3 ( 2)]} = (3,6) Multiplying by the scalar on the right, we get ( 1, 2) ( 3) = {[ 1 ( 3)],[ 2 ( 3)]} = (3,6) This result is shown as a dashed, gray, arrowed line segment pointing in the opposite direction from the original vector, and 3 times as long
Polar vector times positive scalar Imagine some vector a in the polar-coordinate plane whose direction angle is qa and whose magnitude is ra If it s in standard form, we can express it as the ordered pair
a = (qa,ra) When we multiply a on the left by a positive scalar k+, the angle remains the same, but the magnitude becomes k+ra This gives us the left-hand polar product of k+ and a, which is k+a = (qa,k+ra) If we multiply a on the right by k+, we get the right-hand polar product of a and k+, which is a k+ = (qa,rak+)