The following discussion is limited to vectors in a two‐dimensional coordinate plane, although the concepts can be extended to higher dimensions.

If vector **standard position**. If vector

**Figure 1 **

Vectors drawn on a plane.

Vector *P* must be found because point *0* is at the origin. If the coordinates of point A are ( *x* _{a}, *y* _{a} *)* and the coordinates of point *B* are ( *x* _{b}, *y* _{b}), then the coordinates of point P are ( *x* _{b} − *x* _{a}, *y* _{ab} *− y* _{a}).

**Example 1:** If the endpoints of a vector *A*(−2, −7) and B (3, 2), then what are the coordinates of point *P* such that

**Figure 2 **

Drawing for Example 1.

If the coordinates of point *P* are ( *x*, *y*),

An **algebraic vector** is an ordered pair of real numbers. An algebraic vector that corresponds to standard geometric vector *a, b*⟩ if terminal point P has coordinates of *(a, b)*. The numbers *a* and *b* are called the **components** of vector *⟨ a, b⟩* (see Figure 3 ).

**Figure 3 **

Components of a vector.

If *a, b, c*, and *d* are all real numbers such that *a* = *c* and *b* = *d*, then vector **v** = *⟨ a, b⟩* and vector **u** = *⟨ c, d⟩* are said to be equal. That is, algebraic vectors with equal corresponding components are equal. If both components of a vector are equal to zero, the vector is said to be the **zero vector**. The **magnitude** of a vector **v** = *⟨a, b⟩* is

**Example 2:** What is the magnitude of vector **u** = ⟨3, −5⟩?

**Vector addition** is defined as adding corresponding components of vectors—that is, if **v** = *⟨ a, b⟩* and **u** = *⟨c, d⟩*, then **v** + **u** = *⟨a* + *c, b* + *d⟩* (Figure 4 ).

**Figure 4 **

Vector addition.

**Scalar multiplication** is defined as multiplying each component by a constant—that is, if **v** = *⟨a, b⟩* and *q* is a constant, then *q* **v** = *q⟨a, b⟩ = ⟨qa, qb⟩*.

**Example 3:** If **v** = ⟨8, −2⟩ and **w** = ⟨3, 7⟩ then find 5 **v** −2 **w**.

A **unit vector** is a vector whose magnitude is 1. A unit vector **v** with the same direction as a nonzero vector **u** can be found as follows:

**Example 4**: Find a unit vector **v** with the same direction as the vector **u** given that **u** = ⟨7, − 1⟩.

Two special unit vectors, **i** = ⟨1, 0⟩ and **j** = ⟨0, 1⟩, can be used to express any vector **v** = *⟨a, b⟩*.

**Example 5:** Write **u** = ⟨5, 3⟩ in terms of the **i** and **j** unit vectors (Figure 5 ).

**Figure 5 **

Drawing for Example 5.

Vectors exhibit algebraic properties similar to those of real numbers (Table 1).

**Example 6:** Find 4 **u** + 5 **v** if **u** = 7 **i** − 3 **j** and **v** = −2 **i** + 5 **j**.

Given two vectors, **u** = *⟨ a, b⟩* = *a* **i** *+ b* **j** and **v** = *⟨c, d⟩* = *c* **i** + *d* **j**, the **dot product**, written as **u**· **v**, is the scalar quantity **u** ˙ **v** = *ac + bd*. If **u, v**, and **w** are vectors and *q* is a real number, then dot products exhibit the following properties:

The last property, **u ˙ v** = | **u**| | **v**| cos α, can be used to find the angle between the two nonzero vectors **u** and **v**. If two vectors are perpendicular to each other and form a 90° angle, they are said to be **orthogonal**. Because cos 90° = 0, the dot product of any two orthogonal vectors is 0.

**Example 7:** Given that **u** = ⟨ *5*, −3⟩ and **v** = ⟨6, 10⟩, show that **u** and **v** are orthogonal by demonstrating that the dot product of **u** and **v** is equal to zero.

**Example 8:** What is the angle between u = ⟨5, −2⟩ and v = ⟨6, 11⟩?

An object is said to be in a state of **static equilibrium** if all the force vectors acting on the object add up to zero.

**Example 9:** A tightrope walker weighing 150 pounds is standing closer to one end of the rope than the other. The shorter length of rope deflects 5° from the horizontal. The longer length of rope deflects 3°. What is the tension on each part of the rope?

Draw a force diagram with all three force vectors in standard position (Figure 6).

**Figure 6 **

Drawing for Example 9.

The sum of the force vectors must be zero for each component.

For the **i** component: − | **u**|cos 5° + | **v**| cos 3° = 0

For the **j** component: | **u**| sin5° + |v| cos 3° − 150 =

Solve these two equations for | **u**| and | **v**|:

Substituting the values for the sines and cosines:

Multiply the first equation by 0.0872 and the second by 0.9962:

Add the two equations and solve for | **v**|:

Substitute and solve for | **u**|: