Adding and scaling vectors (3D)

What is it about?

A vector describes a translation; it is written as a column and drawn as an arrow. Arrows of equal length and direction represent the same vector.

• Addition (tip to tail): attach $\vec b$ to the tip of $\vec a$; the sum is the diagonal of the parallelogram. Components are added.
• Subtraction: $\vec a - \vec b = \vec a + (-\vec b)$; the difference also points from the tip of $\vec b$ to the tip of $\vec a$.
• Scaling: $k \cdot \vec a$ is parallel to $\vec a$ and $|k|$ times as long; negative $k$ reverses the direction.
• Linear combination: all $r \vec a + s \vec b$ lie in one plane (if $\vec a$, $\vec b$ are not parallel).
• Connecting vector: $\overrightarrow{AB} = \vec b - \vec a$ (“tip minus tail”), midpoint $M = \frac12(A + B)$, length $\sqrt{v_1^2 + v_2^2 + v_3^2}$, e.g. $|(2, 3, 6)| = 7$.
• Find a linear combination: set $r$, $s$ (or drag the tip in the plane) so that $r \vec a + s \vec b = \vec c$. If $\vec c$ is not in the plane, it is impossible.
• Parallelepiped: $\vec a$, $\vec b$, $\vec c$ span it; every corner is a sum of edges, e.g. $\vec a + \vec b + \vec c$. If the three vectors are coplanar, it is flat.
• Unit vector: $\vec v^{\,0} = \vec v / |\vec v|$, e.g. $(1, 2, 2)^T$ has length $3$ and $\vec v^{\,0} = (\frac13, \frac23, \frac23)^T$.
• Centroid: $S = \frac13(A + B + C)$ divides every median $2 : 1$; $T = A + t \cdot \overrightarrow{AB}$ divides $[AB]$ in the ratio $t : (1 - t)$.
• Random walk: the sum of $n$ random unit steps does not depend on their order; on average $|\vec s|^2 = n$.

Try it

1. Play the animation and drag the tips of $\vec a$ and $\vec b$. Load example
2. Set $k$ to $-1.5$. What happens to direction and length? Scaling
3. Rotate the view until you look along the plane of the linear combinations. Linear combination
4. Hit the target $\vec c$. Find
5. Tap corners of the parallelepiped. Parallelepiped
6. Find five integer vectors of length $3$. Length
7. Drag the vertices of the triangle. Centroid
8. Walk 100 steps, then sort them. Random walk

Tasks

Task 1: Sum and difference

Compute $\vec a + \vec b$ and $\vec a - \vec b$ for $\vec a = (2, 4, 0)^T$, $\vec b = (1, -1, 3)^T$. Load task

Task 2: Connecting vector

$A(3, -1, 1)$, $B(-1, 3, 3)$: find $\overrightarrow{AB}$, its length and the midpoint. Load task

Task 3: Centroid

Find the centroid of $A(3, 0, 0)$, $B(0, 3, 0)$, $C(0, 0, 3)$. Load task

Ideas for the classroom

• The shifted copy of $\vec b$ shows that a vector is not tied to a place.
• Common stumbling blocks: computing “tail minus tip”; changing only one sign when subtracting; assuming $|\vec a + \vec b| = |\vec a| + |\vec b|$.