Vectors
Easy Overview
A vector is just a quantity with both size and direction — think of it like an arrow. Speed is a number (scalar), but velocity is a vector because it has direction. This chapter is about adding, subtracting, and multiplying these arrows. Useful for physics, 3D graphics, and navigation.
What's a Vector?
A vector has magnitude (length) and direction. Write it as AB (from A to B) or as a position vector like 2i + 3j. The components tell you how much in each direction. i is the x-direction unit, j is the y-direction, k is the z-direction. Think of them like GPS coordinates — how far east, north, and up.
Vector Addition — The Triangle Law
To add two vectors, put the tail of the second at the head of the first. The result (resultant) goes from the first tail to the second head. It's like walking 5 steps east, then 3 steps north — you end up somewhere northeast. The order doesn't matter: A + B = B + A.
Dot Product (Scalar Product)
The dot product A·B = |A||B|cos θ. It gives you a number, not a vector. It tells you how much one vector goes in the direction of another. If they're perpendicular, dot = 0. If they're parallel, dot = |A||B|. In components: a₁b₁ + a₂b₂ + a₃b₃. Used everywhere in physics for work done.
Cross Product (Vector Product)
The cross product A × B = |A||B|sin θ n̂, where n̂ is perpendicular to both. The result is a VECTOR. If two vectors are parallel, cross = 0. Direction is given by the right-hand rule — curl your fingers from A to B, thumb points the way. In components, it's the determinant of a 3×3 matrix.
Scalar Triple Product
The scalar triple product [A B C] = A·(B × C). It's a number that tells you the volume of the parallelepiped formed by the three vectors. If it's zero, the vectors are coplanar (flat on a plane). Think of it as the '3Dness' check — are these three vectors truly 3D or just pretending?
Key Points
- •Vector = magnitude + direction; Scalar = just magnitude
- •Triangle law: AB + BC = AC
- •Dot product A·B = |A||B|cos θ = a₁b₁ + a₂b₂ + a₃b₃
- •A·B = 0 iff vectors are perpendicular
- •Cross product A×B = |A||B|sin θ n̂ — direction via right-hand rule
- •A×B = 0 iff vectors are parallel
- •Scalar triple product [A B C] = 0 iff vectors are coplanar
Practice Questions
- If A = 2i + 3j − k and B = i − j + 2k, find A·B and A×B.
- Show that vectors A = i + 2j + 3k, B = 2i + 3j − 4k, and C = 3i + 5j − k are coplanar.
- Find the angle between vectors A = i + j + k and B = i − j + k.
- If |A| = 3, |B| = 4, and |A×B| = 6, find the angle between A and B.