Some Laws for Vectors Help

Introduction

When it comes to rules, vectors are no more exalted than ordinary numbers. Here are a few so-called laws that all vectors obey.

Multiplication By Scalar

When any vector is multiplied by a real number, also known as a scalar , the vector magnitude (length) is multiplied by that scalar. The direction remains unchanged if the scalar is positive but is reversed if the scalar is negative.

Commutativity Of Addition

When you add two vectors, it does not matter in which order the sum is performed. If a and b are vectors, then

a + b = b + a

Commutativity Of Vector-scalar Multiplication

When a vector is multiplied by a scalar, it does not matter in which order the product is performed. If a is a vector and k is a real number, then

k a = a k

Commutativity Of Dot Product

When the dot product of two vectors is found, it does not matter in which order the vectors are placed. If a and b are vectors, then

a · b = b · a

Negative Commutativity Of Cross Product

The cross product of two vectors reverses direction when the order in which the vectors are “multiplied” is reversed. That is,

b × a = − ( a × b )

Associativity Of Addition

When you add three vectors, it makes no difference how the sum is grouped. If a, b , and c are vectors, then

( a + b ) + c = a + ( b + c )

Associativity Of Vector-scalar Multiplication

Let a be a vector, and let k ₁ and k ₂ be real-number scalars. Then the following equation holds:

k ₁ ( k ₂ a ) = ( k ₁ k ₂ ) a

Distributivity Of Scalar Multiplication Over Scalar Addition

Let a be a vector, and let k ₁ and k ₂ be real-number scalars. Then the following equations hold:

( k ₁ + k ₂ ) a = k ₁ a + k ₂ a

a ( k ₁ + k ₂ ) = a k ₁ + a k ₂ = k ₁ a + k ₂ a 

Distributivity Of Scalar Multiplication Over Vector Addition

Let a and b be vectors, and let k be a real-number scalar. Then the following equations hold:

k ( a + b ) = k a + k b

( a + b ) k = a k + b k = k a + k b

Distributivity Of Dot Product Over Vector Addition

Let a, b , and c be vectors. Then the following equations hold:

a · ( b + c ) = a · b + a · c

( b + c ) · a = b · a + c · a = a · b + a · c

Distributivity Of Cross Product Over Vector Addition

Let a, b , and c be vectors. Then the following equations hold:

a × ( b + c ) = a × b + a × c

( b + c ) × a = b × a + c × a

= −( a × b ) − ( a × c )

= −( a × b + a × c )

Dot Product Of Cross Products

Let a, b, c , and d be vectors. Then the following equation holds:

( a × b ) · ( c × d ) = ( a · c ) ( b · d ) − ( a · d ) ( b · c )

These are only a few examples of the rules vectors universally obey. If you have trouble directly envisioning how these rules work, you are not alone. Some vector concepts are impossible for mortal humans to see with the “mind’s eye.” This is why we have mathematics. Equations and formulas like the ones in this chapter allow us to work with “beasts” that would otherwise forever elude our grasp.

Practice problems for these concepts can be found at:  Equations, Formulas, And Vectors for Physics Practice Test