Vectors

Vectors represent quantities that have both magnitude and direction, forming the building blocks of linear algebra and physics. This topic covers vector addition and scalar multiplication, dot and cross products, projections, and equations of lines and planes in two and three dimensions.

Vectors are used in physics for forces and velocities, in computer graphics for lighting and transformations, in robotics for motion planning, and in machine learning for representing data points.

Practice Tips

• 1The dot product a dot b = |a||b|cos(theta) gives a scalar; use it to find angles between vectors or to test orthogonality (dot product equals zero).
• 2The cross product a cross b gives a vector perpendicular to both a and b; its magnitude equals the area of the parallelogram they span.
• 3For projection of a onto b, use the formula (a dot b / b dot b) * b, and remember this gives a vector, not a scalar.