Vectors are mathematical objects that have both magnitude and direction. In H2 Mathematics, they are typically represented as column vectors or in terms of unit vectors i and j.
To add or subtract vectors, simply add or subtract their corresponding components. For example, if vector a = (x1, y1) and vector b = (x2, y2), then a + b = (x1 + x2, y1 + y2).
The magnitude of a vector (x, y) is found using the formula √(x² + y²). This represents the length of the vector.
The dot product of two vectors a = (x1, y1) and b = (x2, y2) is calculated as a · b = x1x2 + y1y2. It can also be expressed as |a||b|cosθ, where θ is the angle between the vectors.
Use the formula cosθ = (a · b) / (|a||b|), where a · b is the dot product of the vectors, and |a| and |b| are their magnitudes. Then, solve for θ.
The cross product is primarily used in 3D vector problems (though less common in standard H2). It results in a vector perpendicular to both original vectors. Its magnitude is |a||b|sinθ, representing the area of the parallelogram formed by the vectors.
Vectors can define the direction of lines and the normal direction of planes. Use vector equations of lines (r = a + λd) and planes (r · n = a · n) to solve problems involving intersections, distances, and angles.