Adding vectors and using dot and cross products
A vector has both magnitude and direction, unlike a scalar such as mass or temperature, which has magnitude only; displacement, velocity and force are vectors. In three dimensions a vector is written in components as a = a₁î + a₂ĵ + a₃k̂, where î, ĵ and k̂ are unit vectors along the axes, and its magnitude is √(a₁² + a₂² + a₃²). Vectors are added by the triangle or parallelogram law, which in components simply means adding matching components. The unit vector in the direction of a is a ÷ |a|.
The dot product a · b = |a||b| cos θ is a scalar. It also equals a₁b₁ + a₂b₂ + a₃b₃, so it gives the angle between two vectors, and it is zero exactly when they are perpendicular. The cross product a × b is a vector perpendicular to both a and b with magnitude |a||b| sin θ, which equals the area of the parallelogram formed by the two vectors. The cross product is zero for parallel vectors, and a × b = −(b × a), so the order matters. The projection of a on b is (a · b) ÷ |b|.
Why it matters
Vectors are the natural language of physics and engineering: the forces on a bridge, the velocity of an aircraft in a crosswind and the work done by a force, which is a dot product, are all vector calculations. Torque and magnetic force use the cross product. Games and graphics software use vectors to move objects and calculate lighting, and the chapter leads straight into three-dimensional geometry in Class 12.
Key terms
- Unit vector
- A vector of magnitude one that shows a direction only.
- Dot product
- A scalar equal to the product of two magnitudes and the cosine of the angle between them.
- Cross product
- A vector perpendicular to two given vectors, with magnitude equal to the area of their parallelogram.
- Projection
- The length of the shadow of one vector along the direction of another.