We define vector addition and scalar multiplication algebraically and geometrically.

VEC-0030: Vector Arithmetic

Geometry of Scalar Multiplication

The product of vector with a positive scalar , is a vector that points in the same direction as , and whose length is equal to the length of multiplied by . For example, the figure below shows vectors and . The vectors point in the same direction but the magnitude of is twice the magnitude of .

If a vector is multiplied by , the resulting vector is denoted by . It has the same length as vector , but points in the opposite direction.

Algebra of Scalar Multiplication

We know what scalar multiplication accomplishes geometrically. Our goal now is to translate this idea to an algebraic operation.

Consider vector . We will find an algebraic approach for multiplying by .

Consider to be the hypotenuse of a right triangle.

The head of should be the midpoint of the hypotenuse.

From our study of similar triangles in geometry we know that if we drop perpendiculars from the midpoint of the hypotenuse to the two legs of the triangle, the perpendiculars will bisect the legs.

This tells us that to find and components of we must multiply each component of by .

Consider vector It is clear that multiplying the components of by reverses the direction of while preserving its magnitude.

Explorations init:scalarmult and init:negscalarmult give rise to the following definition of scalar multiplication.

If (), then , and we say that and are scalar multiples of each other.

Geometry of Vector Addition

There are two ways to add vectors geometrically.

“Head-to-Tail” Addition Method

Given vectors and , we can find the sum by sliding so as to place its tail at the head of vector . The vector connecting the tail of with the head of is the sum , as shown in the figure below.

This sum can be interpreted as the total displacement that occurs when traveling along the two vectors starting at the tail of and finishing at the head of .

Note that if we place the tail of at the head of instead, the sum vector will be the same as . Thus, addition of vectors is commutative.

Parallelogram Addition Method

Most of the time we deal with vectors in standard position. So all vector tails are located at the origin. This motivates the parallelogram method for adding vectors.

Observe that if we slide vectors and so that their tails are together, the two vectors determine a parallelogram.

Opposite sides of a parallelogram are congruent and parallel.

Applying the “head-to-tail” addition method shows that the sum is the diagonal of the parallelogram determined by and .

Algebra of Vector Addition

We now know how to add vectors geometrically. Our next goal is to translate this idea to an algebraic operation.

In this problem we will find the sum of and .

To use “head-to-tail” addition method, or to construct the side of a parallelogram opposite of , we want to slide so that its tail is at the point . Observe that has a “run” of and a “rise” of . If we start at , go over then up , we will land on .

The sum is shown below.

We see that the components of can be found by adding the components of and .

Exploration init:vectoradd motivates the following definition.

Vector Subtraction

We can find the difference of two vectors by interpreting subtraction as “addition of the opposite”. Thus, Vector subtraction has an interesting geometric interpretation. As shown in the figure below, if is a diagonal of the parallelogram determined by and , the difference is the other diagonal of the same parallelogram.

Properties of Vector Addition and Scalar Multiplication

We will prove Properties item:inversevectoradd and item:distvectoradd. Proofs of the remaining properties are left to the reader.

Proof of Property item:inversevectoradd
For any vector in , let Then .

Proof of Property item:distvectoradd

Practice Problems

The figure below shows vectors and . Sketch each of the following in the same coordinate plane.

Let Find each of the following
The additive inverse of is