WEBVTT

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Here is the problem: Find the dot product and the angle between ⟨1, 0, 1⟩ and ⟨0, 1, 1⟩.

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Dot product of two 3-dimensional vectors.

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Multiply matching components.

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Add the products.

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Find the magnitudes.

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Angle between the vectors.

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Scalar and vector projection of u onto v.

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So the answer is u times v equals 1, angle approximately 60°.

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

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