Vectors in Two and Three Dimensions

A vector is an ordered list of numbers with its own arithmetic

Arrows give the algebra a picture

Displacement measures travel from one point to another

Distance in space comes from the Pythagorean theorem

Let $\mathbf{u} = (4, -2, 5)$ and $\mathbf{v} = (-1, 3, 5)$. Which vector equals $\mathbf{u} + \mathbf{v}$?

A hiker walks from camp at $P_1(2, 1, 4)$ to a ridge point $P_2(6, -3, 1)$, with all coordinates measured in kilometers. What is the displacement vector of the walk?

A weather balloon starts at position $(2, 5, 1)$ and drifts with constant velocity $(3, 0, 2)$, measured in meters per second. Where is the balloon after 4 seconds?

Two drones report positions $P(1, -2, 2)$ and $Q(3, 1, 0)$ in kilometers. How far apart are they?

A robot starts at $(4, 1, 7)$ and moves with constant velocity $(1, 0, -2)$ for 5 seconds. Give its position at the end as three comma-separated numbers, like 1,2,3.

A sphere has equation $x^2 + y^2 + z^2 - 2x + 4y - 8z + 4 = 0$. Complete the squares and give the radius, written as sqrt(n) for the exact value.