AI Tutor
If an airplane flies at 500Β mph500\text{ mph}, that tells you how fast it moves, but not where it will end up. To navigate safely, air traffic controllers need both the speed and the heading β€” both size and direction.

Scalars vs. Vectors

A scalar is a quantity described completely by a single numerical value called its magnitude (size), such as temperature (72∘F72^\circ\text{F}) or mass (15 kg15\text{ kg}).
A vector is a mathematical object that possesses both magnitude and direction, such as a velocity of 500Β mph500\text{ mph} due North.
How do we draw and write a quantity that points somewhere specific?

Geometric Representation and Component Form

Graphically, we represent a vector as a directed line segment with an initial point (tail) where it starts and a terminal point (head) where the arrow points.
A vector is in standard position when its initial point is placed at the origin (0,0)(0, 0).
If a vector starts at P(x1,y1)P(x_1, y_1) and ends at Q(x2,y2)Q(x_2, y_2), its component form v=⟨vx,vy⟩\mathbf{v} = \langle v_x, v_y \rangle tracks horizontal and vertical change: vx=x2βˆ’x1v_x = x_2 - x_1 and vy=y2βˆ’y1v_y = y_2 - y_1.
Once we break a vector into horizontal and vertical components, how do we measure its total length?

Calculating Magnitude