Scalar product

Objective. Use dot products for projection and angle.

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Learn: the key idea

The scalar (dot) product of two vectors a and b is defined as a·b = |a||b|cos θ, where θ is the angle between them, and equivalently computed in component form as the sum of the products of corresponding components. This product is used to find the angle between vectors via cos θ = (a·b)/(|a||b|), to test perpendicularity (a·b=0 means the vectors are perpendicular), and to find the scalar projection of one vector onto another as (a·b)/|b|. For example, if a=(1,2,2) and b=(2,−1,2), then a·b = 2−2+4 = 4.

Worked example

Find the angle between the vectors a = (3,0,4) and b = (1,2,2), using the scalar product, to one decimal place in degrees.

1 $Compute a\cdotb = (3\times1)+(0\times2)+(4\times2) = 3+0+8 = 11$
2 $Find magnitudes: |a|=\sqrt(9+0+16)=\sqrt25=5, |b|=\sqrt(1+4+4)=\sqrt9=3$
3 $cos θ = 11/(5\times3) = 11/15$
4 $θ = arccos(11/15) \approx 42.8°$

Answer: θ ≈ 42.8°

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Common mistake. Students compute the dot product by multiplying the magnitudes without the cosine term, or confuse it with the vector (cross) product, producing a vector instead of a scalar. The dot product always gives a single number, never a vector.

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