Find the dot product for each pair of vectors.
4i, 5i - 9j

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and returns a single scalar value. It is calculated by multiplying the corresponding components of the vectors and then summing those products. For vectors A = (a1, a2) and B = (b1, b2), the dot product is A · B = a1*b1 + a2*b2. This operation is useful in determining the angle between vectors and in projecting one vector onto another.

Vector Components

Vectors can be expressed in terms of their components along the coordinate axes. In a two-dimensional space, a vector can be represented as A = ai + bj, where 'a' is the component along the x-axis and 'b' is the component along the y-axis. Understanding vector components is essential for performing operations like the dot product, as it allows for the systematic multiplication of corresponding components.

Unit Vectors

Unit vectors are vectors with a magnitude of one and are used to indicate direction. In a Cartesian coordinate system, the unit vectors i and j represent the directions along the x-axis and y-axis, respectively. They are crucial in vector operations, including the dot product, as they help in simplifying calculations and understanding the orientation of vectors in space.