**Vectors in the Cartesian Plane**

A *vector* is a mathematical expression for a quantity with two independent properties: *magnitude* and *direction* . Vectors are used to represent physical variables such as displacement, velocity, and acceleration, when such variables have both magnitude and direction.

Conventionally, vectors are denoted by boldface letters of the alphabet. In the *xy* -plane, vectors **a** and **b** can be illustrated as rays from the origin (0,0) to points ( *x* _{a} , *y* _{a} ) and ( *x* _{b} , *y* _{b} ) as shown in Fig. 6-6.

**Magnitude, Direction, and Sum**

**Magnitude**

The magnitude, or length, of a vector **a** , written | **a** | or *a* , can be found in the Cartesian plane by using a distance formula resembling the Pythagorean theorem:

**Direction**

The direction of **a** , written dir **a** , is the angle *θ* _{a} that **a** subtends counterclockwise from the positive *x* axis. This angle is equal to the arctangent of the ratio of *y* _{a} to *x* _{a} :

dir **a** = *θ* _{a} = arctan ( *y* _{a} /x _{a} )

By convention, the following restrictions hold:

0 ≤ *θ* _{a} < 360 for *θ* _{a} in degrees

0 ≤ *θ* _{a} < 2π for *θ* _{a} in radians

**Sum**

The sum of vectors **a** and **b** , where **a** = ( *x* _{a} , *y* _{a} ) and b = ( *x* _{b} , *y* _{b} ), is given by the following formula:

**a** + **b** = [( *x* _{a} + *x* _{b} ), ( *y* _{a} + *y* _{b} )]

This sum can be found geometrically by constructing a parallelogram with the vectors **a** and **b** as adjacent sides; the vector **a + b** is determined by the diagonal of this parallelogram (Fig. 6-6).

**Multiplication By Scalar and Dot Product**

**Multiplication By Scalar**

To multiply a vector by a *scalar* (an ordinary real number), the *x* and *y* components of the vector are both multiplied by that scalar. Multiplication by a scalar is commutative. This means that it doesn’t matter whether the scalar comes before or after the vector in the product. If we have a vector a = ( *x* _{a} , *y* _{a} ) and a scalar *k,* then

*k a* =

*= (*

**a**k*kx*

_{a},

*ky*

_{a})

**Dot Product**

Let **a** = ( *x* _{a} , *y* _{a} ) and **b** = ( *x* _{b} , *y* _{b} ). The *dot product,* also known as the *scalar product* and written **a** · **b** , of vectors **a** and **b** is a real number (that is, a scalar) given by the formula:

**a** · **b** = *x* _{a} *x* _{b} + *y* _{a} *y* _{b}

**Vectors in the Cartesian Plane Practice Problems**

**Practice 1**

What is the sum of **a** = (3,–5) and **b** = (2,6)?

**Solution 1**

Add the *x* and *y* components together independently:

**Practice 2**

What is the dot product of **a** = (3, –5) and **b** = (2,6)?

**Solution 2**

Use the formula given above for the dot product:

**Practice 3**

What happens if the order of the dot product is reversed? Does the value change?

**Solution 3**

No. The dot product of two vectors does not depend on the order in which the vectors are “dot-multiplied.” This can be proven in the general case using the formula above. Let **a** = ( *x* _{a} , *y* _{a} ) and **b** = ( *x* _{b} , *y* _{b} ). First consider the dot product of **a** and **b** (pronounced “ **a** dot **b** ”):

**a** · **b** = *x* _{a} *x* _{b} + *y* _{a} *y* _{b}

Now consider the dot product **b** · **a** :

**b** · **a** = *x* _{b} *x* _{a} + *y* _{b} *y* _{a}

Because multiplication is commutative for all real numbers, the above formula is equivalent to:

**b** · **a** = *x* _{a} *x* _{b} + *y* _{a} *y* _{b}

But *x* _{a} *x* _{b} + *y* _{a} *y* _{b} is the expansion of **a** · **b** . Therefore, for any two vectors **a** and **b** , it is always true that **a** · **b** = **b** · **a** .

Practice problems for these concepts can be found at: Three-Space and Vectors Practice Test

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