Add, subtract and scale vectors

Add, subtract and scale vectors

Draw the arrows before calculating. Join them head to tail to add, reverse the second arrow to subtract, and check the sign of a scalar before scaling. Moving a free arrow is allowed; rotating it changes the vector.

For nonzero vectors, equality means the same magnitude and direction, regardless of where their free-vector arrows are drawn. In the source grid, a,b,c have equal displacements even though their initial points differ. Zero equals zero too, without needing a direction.

The triangle law places b’s tail at a’s head; a+b joins the first tail to the final head. The source illustrates nonparallel vectors, but addition also works for parallel vectors and zero.

Commutative law:

$$\mathbf a+\mathbf b=\mathbf b+\mathbf a$$

The two boundary routes around the source parallelogram reach the same point and give the same resultant diagonal.

Associative law:

$$\mathbf a+\mathbf b+\mathbf c=(\mathbf a+\mathbf b)+\mathbf c=\mathbf a+(\mathbf b+\mathbf c).$$

The source figure uses the intermediate diagonals a+b and b+c to show two groupings of the same three-leg path.

Subtraction:

$$\mathbf a-\mathbf b=\mathbf a+(-\mathbf b)$$

Reverse b and then apply the triangle law. Its original direction and the reversed direction must remain distinct.

Scalar multiplication: for real $\lambda$,

$$\lvert\lambda\mathbf a\rvert=\lvert\lambda\rvert\lvert\mathbf a\rvert.$$

For nonzero a and nonzero λ, the result is parallel to a. Positive λ keeps the direction and negative λ reverses it; λ=0 produces zero, which has no direction. The source’s parallelism sentence must be read with this zero exception. Its three arrows illustrate a, positive scaling and negative scaling.

Apply the scalar laws

For real scalars m,n and vectors a,b, all four printed properties are:

1. $m\mathbf a=\mathbf a m$ (the scalar may be written on either side; this is scalar multiplication).
2. $(mn)\mathbf a=m(n\mathbf a)=n(m\mathbf a)$ (associativity of scalar multiplication).
3. $(m+n)\mathbf a=m\mathbf a+n\mathbf a$ (distributivity over scalar addition).
4. $m(\mathbf a+\mathbf b)=m\mathbf a+m\mathbf b$ (distributivity over vector addition).

A fixed walking distance multiplies a unit direction, not an arbitrary direction vector. Use the linked Alice question to distinguish $5\widehat{\mathbf p}+3\widehat{\mathbf q}$ from $5\mathbf p+3\mathbf q$. The midpoint proof applies these same path and scaling laws to a triangle.