Adding and Subtracting Vectors

Try this

“Head to tail”: drag the head of $a$ . The vectors $b$ and $c$ after it are carried along, but their lengths and directions stay the same. Does the green sum change?
Make the green sum the zero vector. What do the three arrows form then?
“Parallelogram”: $a$ and $b$ start at the same point, and the diagonal is $a + b$ . Is it the same as the head-to-tail result?
“Subtraction”: $a - b$ goes from the head of which arrow to the head of which?
“Scalar multiple”: drag $λ$ past $0$ . How does the direction of $λ a$ change?

A vector is a quantity with both size and direction. Its size is its modulus (length) $∣ a ∣$ . A vector of length $0$ is the zero vector $0$ ; a vector of length $1$ is a unit vector. Two vectors with the same length and the same direction are equal, wherever they are drawn, so vectors may be slid around freely. There are three operations:

Addition (triangle rule): $A B + B C = A C$ . Put them head to tail; the sum goes from the first starting point to the last head. For several vectors this becomes the polygon rule.
Subtraction: $a - b = a + (- b)$ ; with a common starting point, $O A - O B = B A$ .
Scalar multiplication: $λ a$ has length $∣ λ ∣∣ a ∣$ ; for $λ > 0$ it points the same way, for $λ < 0$ the opposite way, and for $λ = 0$ it is $0$ .

This gives the collinearity theorem: if $a \neq = 0$ , then $b ∥ a ⟺$ there is a unique real number $λ$ with $b = λ a$ .

Predict first

Simplify $A B - A C + B D - C D$ .

0

A D

2 B D

B C

Choose an answer first; then you can see the result.

Why: the centroid in vectors

$G$ is the centroid of $△ A B C$ . Below, the two pictures of addition prove $G A + GB + GC = 0$ . The figure takes $A (- 3, - 2)$ , $B (3, - 2)$ , $C (0, 4)$ , so the centroid $G$ is the origin, and the three vectors are drawn from the same point $G$ .

Proof: GA + GB + GC = 0

Step 1 / 4

Add two first: the parallelogram
$G A + GB$ is the diagonal of the parallelogram with these two as sides. The diagonals bisect each other, so it equals $2 GM$ , where $M$ is the midpoint of $A B$ .

$G A + GB = 2 GM$
Use what the centroid is
All three head to tail
Any other point fails

The general formula in the last line comes from $P A = P G + G A$ : add the three such equations, and the part $G A + GB + GC$ is $0$ .

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: every vector parallel to $a$ can be written as $λ a$

When $a \neq = 0$ , as $λ$ varies, $λ a$ sweeps out the whole line (the gray dashed line): every vector on the line is reached, and $λ$ is unique.

Practice

Exercise 1Basic

Simplify $A B + B C + C D - A D$ . (Enter 0 for the zero vector.)

Exercise 2Basic

$a = (1, 1)$ , $b = (1, 2)$ . Find $∣2 a + b ∣$ .

Exercise 3Advanced

$a = (1, 2)$ , $b = (x, 4)$ , and $a ∥ b$ . Find $x$ .

Exercise 4Challenge

In $△ A B C$ , the point $D$ is on side $B C$ with $B D = 2 D C$ . If $A D = x A B + y A C$ , find $x$ .

Try this

Why: the centroid in vectors

Proof: GA + GB + GC = 0

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Subtraction pointing the wrong way

Can lengths simply be added?