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Dot product is taught in the note for Vectors. Read it below and look for the section on this topic.

Vectors

16 min read

Vectors in three dimensions

In three dimensions a vector is a = a₁i + a₂j + a₃k, with magnitude |a| = √(a₁² + a₂² + a₃²). The ratio theorem states that the point P dividing AB in the ratio m : n (internally) has position vector p = (n a + m b) ÷ (m + n).

Adding vectorsTwo vectors drawn head to tail. Vector A is 4 units to the right and vector B is 3 units upwards. The resultant is the straight line from the start of A to the end of B, which is 5 units long. A parallelogram gives the same result.A = 4B = 3resultant RR = √(A² + B²) = √(16 + 9) = 5parallelogram method

The dot (scalar) product

The dot product is a scalar. It is zero if the vectors are perpendicular.
Worked exampleThe angle between vectors

Find the angle between a = (1, 2, 2) and b = (2, 0, 1).

  1. Dot product.
  2. Magnitudes 3 and √5.

Answer: θ ≈ 53.4°.

The cross (vector) product

The cross product is a vector perpendicular to both a and b. Its magnitude is the area of the parallelogram on a and b.
Worked exampleA cross product

Find a × b for a = (1, 0, 0) and b = (0, 1, 0).

  1. Evaluate the determinant.

Answer: a × b = (0, 0, 1): i × j = k. Note b × a = −k.

Vector differentiation and integration

If r(t) = x(t)i + y(t)j + z(t)k is the position of a particle, then the derivative dr/dt = x′i + y′j + z′k is the velocity. Differentiate and integrate each component separately. For example if r = t²i + 3tj then dr/dt = 2ti + 3j and d²r/dt² = 2i.

Try itDot and cross products

Choose two 3D vectors.

Check yourself The dot product of perpendicular vectors is
Check yourself i × j =
Check yourself (1, 2, 3) · (4, −5, 6) =

Key terms

Dot product
A scalar product a · b = |a||b| cos θ.
Cross product
A vector product perpendicular to both vectors.
Ratio theorem
Gives the position vector of a dividing point.
Scalar
A quantity with only size.

Summary

  • The dot product gives a scalar and the angle between vectors.
  • The cross product gives a perpendicular vector whose size is the parallelogram area.
  • The ratio theorem finds a dividing point.
  • Vectors are differentiated and integrated component by component.
Ratio theorems Cross product