vectors2 - Vectors - Part 2: Products Dot Product (scalar...

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Vectors - Part 2: Products Dot Product (scalar product) If the angle between two vectors A r and B r is θ , then we define the dot product cos( ) AB AB θ ⋅= r r This product is a scalar. The angle between two vectors is always between 0 and 180°. This product obeys the commutative and distributive rules. Special cases: θ = 0 (parallel vectors): A BA B r r (the product of the magnitudes, positive) θ = 90° (perpendicular): 0 AB r r θ = 180° (anti-parallel): A B r r (negative) 2 A AA rr (any vector, dotted with itself, = the magnitude squared) Applied to unit vectors: ˆˆ ˆˆ ˆˆ 1 ii jj kk ⋅=⋅=⋅= ˆˆ ˆ ˆ 0 ij jk k i This makes it easy to find the dot product of vectors in component form. For example, () ( ) ˆ ˆ 23 5 2 5 3 1 7 iji j −⋅+ = = One use of the dot product is to find the angle between two vectors. From the definition above, the angle θ between two vectors is found from cos( ) AB = r r
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As an example, let’s find the angle between the face diagonal and body diagonal of a cube. Let the cube have side = s with the edges aligned in a Cartesian CS.
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vectors2 - Vectors - Part 2: Products Dot Product (scalar...

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