Textbook Question
In Exercises 9–16, letu = 2i - j, v = 3i + j, and w = i + 4j.Find each specified scalar.4(u ⋅ v)
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8. Vectors
Dot Product
Multiple Choice
Which of the following best defines the dot product of two vectors and ?
A
It is the sum of the magnitudes of and :
B
It is the product of the magnitudes of and and the sine of the angle between them:
C
It is the product of the magnitudes of and and the cosine of the angle between them:
D
It is the cross product of and :
Verified step by step guidance1
Recall the definition of the dot product (also called the scalar product) of two vectors \( \mathbf{a} \) and \( \mathbf{b} \). It is a scalar quantity that combines both the magnitudes of the vectors and the angle between them.
The dot product is given by the formula:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \times |\mathbf{b}| \times \cos(\theta) \]
where \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes (lengths) of vectors \( \mathbf{a} \) and \( \mathbf{b} \), and \( \theta \) is the angle between them.
Understand why cosine is used: the dot product measures how much one vector extends in the direction of the other. The cosine function relates the angle to this projection, with \( \cos(0) = 1 \) meaning vectors point in the same direction and \( \cos(90^\circ) = 0 \) meaning they are perpendicular.
Note that the dot product is not the sum of magnitudes, nor does it involve the sine of the angle (which is related to the cross product). Also, the dot product is a scalar, not a vector, so it is not the cross product.
Therefore, the best definition of the dot product is the product of the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) and the cosine of the angle between them, as expressed in the formula above.
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