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).
The dot (scalar) product
Worked exampleThe angle between vectors
Find the angle between a = (1, 2, 2) and b = (2, 0, 1).
- Dot product.
- Magnitudes 3 and √5.
Answer: θ ≈ 53.4°.
The cross (vector) product
Worked exampleA cross product
Find a × b for a = (1, 0, 0) and b = (0, 1, 0).
- 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.
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.