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Plane Vectors Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

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.

Three-Space and Vectors Vectors in the Cartesian Plane

Fig. 6-6. Two vectors in the Cartesian plane. They are added using the “parallelogram method.”

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:

Three-Space and Vectors Vectors in the Cartesian Plane Magnitude

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:

Three-Space and Vectors Vectors in the Cartesian Plane Dot Product

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:

Three-Space and Vectors Vectors in the Cartesian Plane 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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