2.1 Vectors in the plane  (Page 5/29)

To find the magnitude of a vector, we calculate the distance between its initial point and its terminal point. The magnitude    of vector $\text{v}=⟨x,y⟩$ is denoted $\Vert \text{v}\Vert ,$ or $\left|\text{v}\right|,$ and can be computed using the formula

Note that because this vector is written in component form, it is equivalent to a vector in standard position, with its initial point at the origin and terminal point $\left(x,y\right).$ Thus, it suffices to calculate the magnitude of the vector in standard position. Using the distance formula to calculate the distance between initial point $\left(0,0\right)$ and terminal point $\left(x,y\right),$ we have

Based on this formula, it is clear that for any vector $\text{v},$ $\Vert \text{v}\Vert \ge 0,$ and $\Vert \text{v}\Vert =0$ if and only if $v=0.$

The magnitude of a vector can also be derived using the Pythagorean theorem, as in the following figure.

We have defined scalar multiplication and vector addition geometrically. Expressing vectors in component form allows us to perform these same operations algebraically.

Now that we have established the basic rules of vector arithmetic, we can state the properties of vector operations. We will prove two of these properties. The others can be proved in a similar manner.

Proof of commutative property

Let $\text{u}=⟨x_1,y_1⟩$ and $\text{v}=⟨x_2,y_2⟩.$ Apply the commutative property for real numbers:

□

Proof of distributive property

Apply the distributive property for real numbers: