WEBVTT

1
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Here is the problem: Find the dot product and the angle between ⟨1, 2, 3⟩ and ⟨4, negative 5, 6⟩.

2
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Dot product of two 3-dimensional vectors.

3
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Multiply matching components.

4
00:00:13.775 --> 00:00:14.890
Add the products.

5
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Find the magnitudes.

6
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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 12, angle approximately 68.562447°.

9
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The dot product is positive, so the angle between the vectors is acute.

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