Vectors give geometry a way to describe both size and direction, not just location. A vector can represent a move from point A to point B on a coordinate plane, such as 3 units right and 2 units up. This makes vectors useful for connecting algebra, geometry, and motion in one visual idea.
They matter because many transformations, paths, forces, and velocities can be modeled with arrows.
Key Facts
- A vector from A(x1, y1) to B(x2, y2) is <x2 - x1, y2 - y1>.
- The magnitude of v = <a, b> is |v| = sqrt(a^2 + b^2).
- Vector addition is <a, b> + <c, d> = <a + c, b + d>.
- A translation by vector <a, b> moves point (x, y) to (x + a, y + b).
- The opposite vector of <a, b> is <-a, -b>, which has the same magnitude but opposite direction.
- A unit vector in the direction of v is v/|v|, so for v = <a, b> it is <a/|v|, b/|v|>.
Vocabulary
- Vector
- A vector is a quantity with both magnitude and direction, often drawn as an arrow.
- Component
- A component is one part of a vector along an axis, such as the horizontal or vertical part.
- Magnitude
- Magnitude is the length or size of a vector.
- Translation
- A translation is a rigid motion that moves every point of a figure the same distance in the same direction.
- Resultant
- The resultant is the single vector produced by adding two or more vectors.
Common Mistakes to Avoid
- Subtracting coordinates in the wrong order: the vector from A to B must be <x2 - x1, y2 - y1>, not <x1 - x2, y1 - y2>.
- Adding magnitudes instead of components: vectors must be added by combining horizontal parts and vertical parts separately.
- Treating a vector as a fixed point: a vector can be moved without changing it as long as its length and direction stay the same.
- Forgetting direction when using magnitude: |v| gives only the length, so vectors with the same magnitude can point in different directions.
Practice Questions
- 1 Point A is (2, -1) and point B is (7, 3). Find the vector from A to B and its magnitude.
- 2 Add the vectors u = <4, -2> and v = <-1, 5>. Then use the result to translate the point P(3, 1).
- 3 A triangle is translated by the vector <6, -3>. Explain how each vertex changes and why the triangle keeps the same size and shape.