A scalar has magnitude only. A vector also has a direction. Temperature is a scalar; displacement is a vector. Rotating a vector can preserve its magnitude while changing its components. The unit vectors î and ĵ select the positive coordinate directions.
Move the tail of the second arrow to the head of the first. The resultant joins the original tail to the final head. Subtracting B means adding −B. A negative scale factor reverses direction; its absolute value scales length. Equal vectors have equal components, regardless of where they are drawn.
Rectangular components are the projections of a vector onto perpendicular axes. With θ measured from +x, the adjacent projection is F cos θ and the opposite projection is F sin θ. The force equals the vector sum of those components. Rotating through a quadrant changes component signs.
The dot product is a scalar measuring alignment: positive for acute angles, zero at 90°, negative for obtuse angles. The cross product is a vector perpendicular to both inputs. Its z-component is signed in this plane. The right-hand rule gives its direction, and reversing the order reverses that direction.
Uniform motion in a plane has constant horizontal and vertical velocity components. Each coordinate evolves as a separate straight-line equation, but both use the same elapsed time. Combining the coordinates gives a straight path. Changing one component changes the path direction and speed.
Resolve the launch velocity into horizontal and vertical components. With no air resistance, horizontal velocity stays constant while vertical velocity decreases by g each second. At the apex the vertical velocity is zero; horizontal velocity usually remains. The ball follows a parabola until ground contact.
Velocity includes direction. On a circle its magnitude stays constant but its direction continually changes. The velocity is tangent to the circle, while acceleration points inward. The two are perpendicular. Doubling speed quadruples the centripetal acceleration at the same radius.