Objectives
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In this article, we will look at another representation of vectors, as well as the basics of vector multiplication.
Unit Vectors
Although the coordinate form for representing vectors is clear, we can also represent them as algebraic expressions using unit vectors. In our standard rectangular (or Euclidean) coordinates (x, y, and z), a unit vector is a vector of length 1 that is parallel to one of the axes. In the two-dimensional coordinate plane, the unit vectors are often called i and j, as shown in the graph below. For three dimensions, we add the unit vetor k corresponding to the direction of the z-axis. These vectors are defined algebraically as follows.
i = (1, 0) or (1, 0, 0)
j = (0, 1) or (0, 1, 0)
k = (0, 0, 1)
Before we present an algebraic representation of vectors using unit vectors, we must first introduce vector multiplication--in this case, by scalars.
Vector Multiplication by Scalars
Multiplication involving vectors is more complicated than that for just scalars, so we must treat the subject carefully. Let's start with the simplest case: multiplying a vector by a scalar. Below is the definition for multiplying a scalar c by a vector a, where a = (x, y). (Again, we can easily extend these principles to three dimensions.)
Scalar multiplication is commutative, so . But what does this multiplication mean? As it turns out, multiplication by a scalar c has the effect of extending the vector's length by the factor c. This is most clearly seen with unit vectors, but it applies to any vector. (Multiplication by a negative scalar reverses the direction of the vector, however.) The graph below shows some examples using c = 2. (Recall that the location of a vector doesn't affect its value.)
Practice Problem: Given a vector a = (3, 1), find a vector in the same direction as a but twice its length.
Solution: When we multiply a vector by a scalar, the direction of the product vector is the same as that of the factor. The only difference is the length is multiplied by the scalar. So, to get a vector that is twice the length of a but in the same direction as a, simply multiply by 2.
Chatty for facebook 2 1 – lightweight facebook chat. 2a = 2 • (3, 1) = (2 • 3, 2 • 1) = (6, 2)
Algebraic Representation of Vectors
We can use scalar multiplication with vectors to represent vectors algebraically. Note that any two-dimensional vector v can be represented as the sum of a length times the unit vector i and another length times the unit vector j. For instance, consider the vector (2, 4). Apply the rules of vectors that we have learned so far:
(2, 4) = (2, 0) + (0, 4) (addition rule for vectors)
(2, 4) = 2 • (1, 0) + 4 • (0, 1) (multiplication rule for scalars and vectors)
(2, 4) = 2i + 4j
Graphically, we are adding two vectors in the unit directions to get our arbitrary vector.
Note that the unit vectors act almost identically to variables. Thus, we can add two vectors a and b as follows.
a = 3i – 2j b = i + 3j
a + b = (3i – 2j) + (i + 3j) = 3i + i – 2j + 3j = 4i + j
This representation provides more flexibility than the coordinate representation, but it is equivalent.
Practice Problem: Calculate the sum and difference (t - u) of the vectors t = -2i + 3j and u = 6i - 4j.
Solution: We can solve this problem algebraically quite easily.
t + u = (-2i + 3j) + (6i - 4j) = 4i - j = (4, -1)
t - u = (-2i + 3j) - (6i - 4j) = -2i + 3j - 6i + 4j = -8i + 7j = (-8, 7)
Vector Multiplication: The Scalar (Dot) Product
Multiplication of two vectors is a little more complicated than scalar multiplication. Two types of multiplication involving two vectors are defined: the so-called scalar product (or 'dot product') and the so-called vector product (or 'cross product'). For simplicity, we will only address the scalar product, but at this point, you should have a sufficient mathematical foundation to understand the vector product as well. The scalar product (or dot product) of two vectors is defined as follows in two dimensions. As always, this definition can be easily extended to three dimensions-simply follow the pattern. Note that the operation should always be indicated with a dot (•) to differentiate from the vector product, which uses a times symbol ()--hence the names dot product and cross product.
The meaning of this product may not be entirely clear to you at this point, however. We can illustrate by looking at a simple case: the scalar product of an arbitrary vector v and the unit vectors i and j.
Thus, v • i is the 'part' of vector v in the direction of I.
This explanation only works, however, for vectors of length 1. When two arbitrary vectors are multiplied, the scalar product has a similar meaning, but the magnitude of the number is a little different. We won't go into this in any further depth, but we can consider a special case where the scalar product yields valuable information.
The Length of a Vector
Consider the case of a scalar product of a vector v with itself.
Let's look at this situation graphically. Vidconvert 1 4 5.
The result here is a right triangle with a horizontal leg of length x and a vertical leg of length y. These lengths correspond to the lengths of the component vectors xi and yj, respectively. But we know from the Pythagorean theorem that is the square of the length of vector v. Not coincidentally, this is the same as the scalar product of v with itself. Thus, the length of any vector v, written as (or sometimes ) is the square root of the scalar product.
In the simple case of the unit vectors,
These simple cases help verify this interpretation of the scalar product.
Practice Problem: Tutorial articles. Calculate the lengths of the following vectors.
a. b. 3i + 2j – k c. (2, –1) d. 5j
Solution: In each case, simply take the square root of the scalar product of the vector with itself. The result is the length of the vector in each case. For part b, simply extend the definition of the scalar product to three dimensions.
a.
b.
c.
d.
Calculator Use
Solve math problems using order of operations like PEMDAS, BEDMAS and BODMAS. (PEMDAS Warning) This calculator solves math equations that add, subtract, multiply and divide positive and negative numbers and exponential numbers. You can also include parentheses and numbers with exponents or roots in your equations.
Use these math symbols:
+ Addition
- Subtraction
* Multiplication
/ Division
^ Exponents (2^5 is 2 raised to the power of 5)
r Roots (2r3 is the 3rd root of 2)
() [] {} Brackets
You can try to copy equations from other printed sources and paste them here and, if they use ÷ for division and × for multiplication, this equation calculator will try to convert them to / and * respectively but in some cases you may need to retype copied and pasted symbols or even full equations.
If your equation has fractional exponents or roots be sure to enclose the fractions in parentheses. For example:
- 5^(2/3) is 5 raised to the 2/3
- 5r(1/4) is the 1/4 root of 5 which is the same as 5 raised to the 4th power
Entering fractions
If you want an entry such as 1/2 to be treated as a fraction then enter it as (1/2). For example, in the equation 4 divided by ½ you must enter it as 4/(1/2). Then the division 1/2 = 0.5 is performed first and 4/0.5 = 8 is performed last. If you incorrectly enter it as 4/1/2 then it is solved 4/1 = 4 first then 4/2 = 2 last. 2 is a wrong answer. 8 was the correct answer.
Math Order of Operations - PEMDAS, BEDMAS, BODMAS
PEMDAS is an acronym that may help you remember order of operations for solving math equations. PEMDAS is typcially expanded into the phrase, 'Please Excuse My Dear Aunt Sally.' The first letter of each word in the phrase creates the PEMDAS acronym. Solve math problems with the standard mathematical order of operations, working left to right:
- Parentheses - working left to right in the equation, find and solve expressions in parentheses first; if you have nested parentheses then work from the innermost to outermost
- Exponents and Roots - working left to right in the equation, calculate all exponential and root expressions second
- Multiplication and Division - next, solve both multiplication AND division expressions at the same time, working left to right in the equation.
- Addition and Subtraction - next, solve both addition AND subtraction expressions at the same time, working left to right in the equation
PEMDAS Warning
Multiplication DOES NOT always get performed before Division. Multiplication and Division happen at the same time, from left to right.
Addition DOES NOT always get performed before Subtraction. Addition and Subtraction happen at the same time, from left to right.
The order 'MD' (DM in BEDMAS) is sometimes confused to mean that Multiplication happens before Division (or vice versa). However, multiplication and division have the same precedence. In other words, multiplication and division are performed during the same step from left to right. For example, 4/2*2 = 4 and 4/2*2 does not equal 1.
The same confusion can also happen with 'AS' however, addition and subtraction also have the same precedence and are performed during the same step from left to right. For example, 5 - 3 + 2 = 4 and 5 - 3 + 2 does not equal 0.
A way to remember this could be to write PEMDAS as PE(MD)(AS) or BEDMAS as BE(DM)(AS).
Order of Operations Acronyms
The acronyms for order of operations mean you should solve equations in this order always working left to right in your equation.
PEMDAS stands for 'Parentheses, Exponents, Multiplication andDivision, Addition andSubtraction'
You may also see BEDMAS and BODMAS as order of operations acronyms. In these acronyms, 'brackets' are the same as parentheses, and 'order' is the same as exponents.
BEDMAS stands for 'Brackets, Exponents, Division andMultiplication, Addition andSubtraction'
BEDMAS is similar to BODMAS.
BODMAS stands for 'Brackets, Order, Division andMultiplication, Addition andSubtraction'
Operator Associativity
Multiplication, division, addition and subtraction are left-associative. This means that when you are solving multiplication and division expressions you proceed from the left side of your equation to the right. Similarly, when you are solving addition and subtraction expressions you proceed from left to right.
Examples of left-associativity:
- a / b * c = (a / b) * c
- a + b - c = (a + b) - c
Exponents and roots or radicals are right-associative and are solved from right to left.
Examples of right-associativity:
- 2^3^4^5 = 2^(3^(4^5))
- 2r3^(4/5) = 2r(3^(4/5))
For nested parentheses or brackets, solve the innermost parentheses or bracket expressions first and work toward the outermost parentheses. For each expression within parentheses, follow the rest of the PEMDAS order: First calculate exponents and radicals, then multiplication and division, and finally addition and subtraction.
You can solve multiplication and division during the same step in the math problem: after solving for parentheses, exponents and radicals and before adding and subtracting. Proceed from left to right for multiplication and division. Solve addition and subtraction last after parentheses, exponents, roots and multiplying/dividing. Again, proceed from left to right for adding and subtracting.
Adding, Subtracting, Multiplying and Dividing Positive and Negative Numbers
This calculator follows standard rules to solve equations.
Rules for Addition Operations (+)
If signs are the same then keep the sign and add the numbers.
If signs are different then subtract the smaller number from the larger number and keep the sign of the larger number.
Rules for Subtraction Operations (-)
Keep the sign of the first number. Change all the following subtraction signs to addition signs. Change the sign of each number that follows so that positive becomes negative, and negative becomes positive then follow the rules for addition problems.
Rules for Multiplication Operations (* or ×)
Multiplying a negative by a negative or a positive by a positive produces a positive result. Multiplying a positive by a negative or a negative by a positive produces a negative result.
Gifox 2 0 2 Multiplication
Rules for Division Operations (/ or ÷)
Gifox 2 0 2 Multiplication Worksheets
Similar to multiplication, dividing a negative by a negative or a positive by a positive produces a positive result. Dividing a positive by a negative or a negative by a positive produces a negative result.
Gifox 2 0 2 Multiplication Tables
