James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 12.3 The Dot Product 809
ExamplE 2 If the vectors a and b have lengths 4 and 6, and the angle between them is
y3, find a ? b.
SOLUtion Using Theorem 3, we have
a ? b − | a | | b | cossy3d − 4 ? 6 ? 1 2 − 12
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The formula in Theorem 3 also enables us to find the angle between two vectors.
6 Corollary If is the angle between the nonzero vectors a and b, then
cos −
a ? b
| a | | b |
ExamplE 3 Find the angle between the vectors a − k2, 2, 21l and b − k5, 23, 2 l.
SOLUtion Since
| a | − s2 2 1 2 2 1 s21d 2 − 3 and | b | − s5 2 1 s23d 2 1 2 2 − s38
and since
we have, from Corollary 6,
So the angle between a and b is
a ? b − 2s5d 1 2s23d 1 s21ds2d − 2
cos −
a ? b
| a | | b | − 2
3s38
− cos 21S 2
3s38
D < 1.46 sor 84°d
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Two nonzero vectors a and b are called perpendicular or orthogonal if the angle
between them is − y2. Then Theorem 3 gives
a ? b − | a | | b | cossy2d − 0
and conversely if a ? b − 0, then cos − 0, so − y2. The zero vector 0 is considered
to be perpendicular to all vectors. Therefore we have the following method for determining
whether two vectors are orthogonal.
7 Two vectors a and b are orthogonal if and only if a ? b − 0.
ExamplE 4 Show that 2i 1 2j 2 k is perpendicular to 5i 2 4j 1 2k.
SOLUtion Since
s2i 1 2j 2 kd ? s5i 2 4j 1 2kd − 2s5d 1 2s24d 1 s21ds2d − 0
these vectors are perpendicular by (7).
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