WEBVTT

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Here is the problem: Find the dot product and the angle between ⟨5, 2, negative 1⟩ and ⟨1, negative 3, 4⟩.

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Dot product of two 3-dimensional vectors.

3
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Multiply matching components.

4
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Add the products.

5
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Find the magnitudes.

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Angle between the vectors.

7
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Scalar and vector projection of u onto v.

8
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So the answer is u times v equals negative 5, angle approximately 100.31319°.

9
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The dot product is negative, so the angle between the vectors is obtuse.

10
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Computed exactly; the dot product is symmetric, so v times u gives the same value.