Categories
Algebra

Is Linear Algebra Hard?

Is Linear Algebra Hard? Technically Not But It Can Feel Hard

Linear algebra is about the properties of vector spaces and matrices and it is, along with calculus, fundamental to all higher mathematics. Calculus is considered the hardest mathematics subject in high school with only a small percentage of students reaching calc.

So the big question, “Is linear algebra hard?” Linear algebra is not hard! Most linear algebra concepts can be visualized by very visual students, such as imagining vectors in three dimensions, who will find it easier to understand than calc. Once you get the hang of writing out matrices, vectors, and other objects, associating them to the basic image of vectors as arrows can be very intuitive.

Some students would say that linear algebra is harder than calculus, read more the find out why!

Is Linear Algebra a Hard Class In College?

Linear algebra, compared to other math classes, is generally seen as being one of the easier college math courses. The difficulty will depend on your prior knowledge in maths, the professor, and college. If you have taken and done well in math classes such as single and multivariable calc, discrete maths, or an intro to proofs, then, linear algebra shouldn’t be a problem. Of course, the professor and caliber of the college will also dictate the pace and difficulty of the assessment tasks or exams. The faster the pace, the more material you need to learn in a shorter time while exams that are more proof-based will be more difficult.

Is Linear Algebra Harder Than Calculus?

The answer to this question is dependent on your situation, for example, a student with a major in maths, physical, or computer science would usually be taught more difficult content than single-variable calc, usually known as calculus 1 and 2.

Linear algebra studies straight lines relating to linear equations while calculus is about consistently changing variables relating to derivatives, integrals, vectors, matrices, and parametric curves. The reason why some people consider linear algebra course harder than calculus is that linear algebra:

  1. Is usually the first course into rigorous proof so it is a new way of thinking
  2. Is less just about computations or performing rote mathematical operations than how calc is taught
  3. Has fewer free materials online to help you go through problems
  4. Is not taught in high school, so many have more exposure to calculus than linear algebra

In the end, it is difficult to conclude the answer to “Is linear algebra harder than calculus?” as that is very subjective. However, the mathematical concepts in linear algebra themselves are not particularly harder than calculus however these are usually newer concepts with fewer free resources available.

In more detail, the following are some factors that may the difficulty of your linear algebra course:

It Depends On Which Department You Take It With

The difficulty of your linear algebra course will depend on the department you take it in, for example, linear algebra within the math department will more likely be more abstract and ask for more proofs in exams which would be more difficult. Meanwhile, the computer science department will more likely focus on the computational nature of linear algebra. linear algebra is more abstract and they ask for more proofs on exams. If you learn linear algebra from the computer science department you may find it more abstract than linear algebra.

It Depends on What You Already Know

Just like calculus, the difficulty of linear algebra will depend on your level of mathematical maturity and previous exposure to solving math proofs. If you haven’t taken many math classes in college yet, it is more likely that you’ll find the classes more challenging but of course, it doesn’t mean you will do badly.

On the other hand, if you have completed and achieved good grades in other lower-division maths courses such as single and multivariable calculus, discrete maths, or an intro course to proofs, you will most likely do well in linear algebra. You may even notice that there are overlaps at the beginning of the multivariable calculus course.

It Will Depend a Lot On The Professor

The difficulty of your college linear algebra class will really depend on your professor. The pace, content, materials, homework, exam, and even guidance on what you need to study are all dictated by your professor.

Since this is usually the first course students are exposed to proofs in college, professors who focus a lot on proofs will make your classes and exams a lot more difficult. A good way to assess the professor is to check professor rating sites, asking your peers/students who have taken this course before, and checking faculty social media groups. Just be fast, the classes with higher-rated professors fill up fast.

It Will Also Depend On The College

For linear algebra, high-caliber colleges or ones with low acceptances rates tend to speed through more content in less time. In this case, classes usually more challenging and require more study time. On the other hand, colleges with higher acceptance rates tend to get through less content in more time so it is likely to be much easier.

However, the college does not guarantee or fully determine the difficulty of your linear algebra class as the professor will still be a major factor.

Linear Algebra Vs Multivariable Calculus

Linear algebra is the study of vectors and allows us to solve systems modeled with multiple linear equations while in multivariable calculus we introduce two or more independent variables to expand single-variable calculus and be able to apply it to a three-dimensional coordinate system.

Tips for succeeding in a linear algebra course

Prioritize What The Professor Gives You

In most cases, materials given to you by the professor will be in the exam so before taking the exam, make sure to prioritize these materials. Sometimes the professor will provide a study guide, be sure to follow it!

Plan Out The Semester As Soon As You Can

Always look through the syllabus for all your courses before your semester starts! Planning will allow you to spot any difficult or busy weeks and gives you an idea of when your exams or homework is due. This way you won’t miss any assessments and know when you need to start studying for any exams.

Make Sure To Do Well On The Homework

Many students do not take homework seriously at all. Make sure to do well on the homework you will boost your grade up. By doing well it will make up for poor exams and score better on those which will be harder to get a top 10. Do well on homework to bump up your grade in linear algebra and make sure you get a good grade.

Get Assistance When You Need It

Get help when you are stuck!

First, put in the effort to find a solution yourself to understand what you don’t get. This will make the solution more satisfying to finally get and it shows to the professor that you are putting an effort into the class.

Sometimes when you still don’t understand or fail to find a solution, sleep on it, you’ll be surprised how well this works.

Read The Textbook Before Jumping To The Problems

Spend time reading through the chapter in your textbook before you attempt the problems. In linear algebra class, the professor may not have time to cover everything that was in the book and it is always good to make sure you read again and ensure understand the fundamental concepts. So don’t jump straight into the homework problems, spending some time understanding the concept will make more advanced concepts easier to comprehend and thus solve.

Always ask why you need to solve the maths the way it is being shown.

Categories
Calculus

Is Calculus Hard to Learn?

Is Calculus Hard To Learn? The Verdict is it Doesn’t Have to Be

Math can be hard for some students but it’s not always hard to learn for others. When you first start learning calculus in high school math class it can look hard. It is hard when you don’t know algebra, trigonometry, and geometry, for example, you need to be fairly good with functions from algebra class.

It is essential to be comfortable with your mathematical foundations before you start learning calculus. This will make the learning process less overwhelming. You may find that coming back to content from previous classes will seem easier!

Students who tend to fail at Calculus see the concepts as meaningless procedures with no function. But it is the language of motion and change. For more information read Infinite Powers: How Calculus Reveals the Secrets of the Universe

What Do Many Students Studying a Calculus Course Say?

Many students say that calculus is not hard particularly when you are okay with algebra, trig, and geometry however it depends on your level of interest and your teacher. Our tip is to read over concepts and ideas before applying derivative and integration procedurally.

Calculus and other advanced mathematics classes can be difficult for all students to learn but with practice, it does get easier.

Calculus in High School

These days calculus is a high school topic for the most academically strong students and it is taught through Advanced Placement (AP) Calculus and there are two versions AB which equates to a semester of college Calculus and BC which equates to a whole year.

Calculus class in high school should only be taken if the student is confident in algebra, trigonometry, and geometry. If not then it is better the student doesn’t rush into calculus and rather strengthen his/her foundations.

Taking calculus in high school is not necessary, for example, it is common that college students who repeat the Calculus course after high school would lose interest in mathematics and along with their confidence.

Calculus in College

At the college level, students are placed in college classes where teachers are not there to hold your hand the whole way through. Motivation needs to come from the student themselves as no one is going to chase you up to do your homework. Another difference is that a calculus course is one semester-long, not a whole year so students get less time to dilly dally.

Calculus is hard in college and many students fail the exam and the cause is mainly due to the lack of interest and having weak foundations in algebra, trigonometry, and geometry. It is okay to fail, just don’t give up.

FAQs

Is Calculus Really That Hard?

Is calculus hard? Yes, many students struggle with calculus but it doesn’t have to be this way.

In terms of skills, it is usually because they’re lacking in the fundamentals in algebra, trigonometry, and geometry but don’t be discouraged. Most students have weaknesses and these can be strengthened with practice. Ideally, you strengthen this topic before you take calculus.

Math is sequential, which means concepts you learn in previous classes will relate to the next class topic. So don’t rush in calculus.

Is Calculus Easier than Algebra?

Calculus is harder than algebra. They’re about the same in terms of difficulty but calculus is more complex, requiring you to draw on everything you learned in geometry, trigonometry, and algebra. Calculus is built on top of these previous courses so without a solid foundation all around, you’re doomed to fail calculus.

What is the Hardest Math Class?

Advanced calc (aka analysis) is the hardest math class in undergraduate and many consider this course to be the rite of passage into graduate studies in mathematics.

In Harvard, this course is called Math 55 and is described by the Harvard University Department of Mathematics as “probably the most difficult undergraduate math class in the country”.

Can the Average Person Learn Calculus?

Is calculus hard? Yes.

Can the average person learn it? Yes!

The average person can learn and do calculus to a high school level with some difficulty. However many would struggle at the college level particularly for engineering, sciences, and mathematics degrees.

Calc class is hard for many people and takes practice to learn to be good at it.

Categories
Algebra

Dot Product Calculator

Dot product calculator

The online vector dot product calculator allows you to find the total dot product of two vectors by multiplying the two vectors with each other.

This online calculator will help you compute:

  • Vector components

=

0

  • Magnitude and angle

a · b =

0 * 0 * cos(0) = 0

Magnitude of a =

Magnitude of b =

Angle θ between vector a and b =

°

The resultant of this calculation i.e. the total dot product is a scalar, not a vector. The dot product finds the total length of the two vectors (length). The i, j, and k fields are multiplied together, and then all values are added up to give the total dot product.

How to use the dot product calculator

The dot product calculator, also known as the dot product of two vectors calculator or matrix dot product calculator, is straightforward to use. Simply enter the required values and use our online calculator to find the total dot product in a few easy steps:

  1. First, input the 3 values for vector a (x, y, z).
  2. Second, input the 3 values for vector b (x, y, z).
  3. Finally, after you've entered all the values, click calculate. You'll automatically get the solution, the dot product of vector c (x, y, z) and the angle between vectors.

Note: You must input at least 2 vector values and the number of terms must be the same for all vectors. You can enter the values in parentheses, square brackets, greater or less than signs or simply begin a new line for each vector. Values entered may be decimals or integers, but cannot be fractions or functions.

What is a dot product?

In linear algebra, a dot product (also known as a scalar product or inner product) is a result of multiplying the numerical values of two or more vectors. The scalar product of two vectors is equal to the product of their magnitudes. Therefore, two perpendicular vectors will have a dot product of zero.

Dot product formula

For vectors a = (a1, a2, a3) and b = (b1, b2, b3)the dot product can be found by using the following formula:

a · b = a1b1 + a2b2 + a3b3

Given that the dot product is the product of the magnitudes of vectors multiplied by the value of the cosine of the angle between the vectors, this can be expressed as the following equation: 

ab = a b cosθ

So, if a vector had 3 components, the dot product formula would be:

a•b = a₁ * b₁ + a₂ * b₂ + a₃ * b₃

Since cosine is the ratio of the vectors’ magnitudes and the scalar products. This can be further expressed as:

cosα = a•b / (|a| * |b|).

where:

  • a and b are two vectors
  • |a| and |b| are the magnitudes of the vector a and b respectively
  • Θ is the angle between the two vectors

The dot product calculator can also be used to solve for the angle between two given vectors. To find the angle Θ between the vectors, use the following equation:

Θ = Cos-1 a.b / |a| |b|

How to calculate the dot product manually

To perform the dot product calculator yourself by hand, you'll need to draw both vectors a and b and separate them with an angle. Note this only works well for 2D vectors.

Alternatively, using the formulas we've provided above, you should now be able to solve for the dot product or angle between two given vectors when given input variables.

Solving dot product with vector component - Example 1

Let's assume we know the input variables for the two vectors a and b.

  • Example: If vector a = [4, 5, -3] and vector b = [1, -2, -2]. How can we find out the dot product of these two vectors?
  • Input the values into the equations.
  • a ⋅ b = (a1 * b1) + (a2 * b2) + (a3 * b3)
  • a ⋅ b = (4 * 1) + (5 * -2) + (-3 * -2)
  • a ⋅ b = 4 -10 +6
  • a ⋅ b = 0

Solving dot product with magnitude and angle - Example 2

In order to solve for the dot product in this example, we need to know the magnitude for both vectors and the angle between the two vectors.

  • example: A vector has magnitude 10 and b vector has magnitude of 15, the angle between the vectors is 60degree. Find the dot product of two vectors?

Solution:

Step 1:

Here,

|a| = 10

|b| = 15

Θ = 60degree

Step 2:

a.b = |a| |b| cosΘ

a.b = 10 * 15 cos60

a.b = 150 cos60

a.b = 150 (0.5)

a.b = 75

FAQs

What is the dot product of a vector with itself?

The dot product of a vector with itself is the square of its magnitude. Since the dot product of two vectors is commutative, the order of the vectors in the product does not matter.

Applications of the dot product

Some applications of the dot product include:

  • determining whether two vectors are perpendicular or parallel to each other
  • proving the law of cosine
  • physical quantities defined as a dot product such as power, electric or magnetic flux and magnetic potential energy
Categories
Algebra

Cross Product Calculator – Cross Product of Two Vectors (Vector Product)

The Cross Product Calculator calculates the cross product of two given vectors, a and b. To find the cross product, enter the x,y, and z values of two vectors into the calculator below to compute the resulting cross product.

X

=

(0, 0, 0)

How to use the cross product calculator

This vector multiplication calculator is simple and easy to use. Instead of manual computations, it provides you with the vector cross product in a matter of seconds. Use our online vector product calculator to find the cross products.

  1. Input the 3 values for vector a (x, y, z).
  2. Input the 3 values for vector b (x, y, z).
  3. After you've entered all the values, you'll get the solution, the cross products of vector c (x, y, z)

Cross product formula

The formula for calculating the new vector of the cross product of two vectors is:

a × b =‖a‖ ‖b‖ sin(θ) n

where:

θ is the angle between a and b in the plane containing them (between 0 – 180 degrees)

a‖ and ‖b‖ are the magnitudes of vectors a and b

n is the unit vector perpendicular to a and b

Vector coordinates

In terms of vector coordinates, the above formula can be simplified as follows:

a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Where:

a and b are vectors with coordinates (a1,a2,a3) and (b1,b2,b3).

The direction of the resulting vector can be determined with the right-hand rule.

The vectors in three dimensions are: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)

From the definition, these rules apply:

i × j = k j × i = i k × i = j

j × i = -k k × j = -i i × k = -j

i × i = j × j = k × k = 0

the cross product of both vectors will be:

a × b = (a1i + a2j + a3k) × (b1i + b2j + b3k)

Example of how to calculate the cross product of two vectors

First, we need two vectors a and b
Given:
vector a has coordinates of (2,1,4) and
vector b has coordinates of (3,1,2).

Second, use the simplified equation above to calculate the resulting vector coordinates of the cross product.

a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

We need to find the x coordinate. Through the formula above, we find x to be 1*2 - 4*1 = 2 - 4 = -2.

Using the same equation, we find y and z to be 8 and -1 respectively.

Finally, we find the new vector from the cross product of a × b is (-2, 8, -1)

Important: Don't forget that in a cross product, the result of a × b is not the same as b × a. In fact a × b = -b × a.

FAQs

What is a cross product?

A cross product is a vector product that is perpendicular to both of the original vectors and is over the same magnitude.

Cross product definition

A cross product (aka a vector product) is the result of a cross product between two vectors, i.e., a new vector that is perpendicular to both vectors. The magnitude of the new vector is equal to the area of a parallelogram with the sides of the two original vectors.

How to calculate the cross product

The multiplication of vectors refers to the cross product, and it can be done in two of the following ways, depending on which input you know.

1. Magnitude of Vector A and B

A cross product tells you how different dimensions are interacting with each other. If you know the magnitude of the vector a and b, then you can compute the magnitude by multiplying them with a sine angle forming between both vectors.

a∥=√(a12+a22+a32)

a × b =‖a‖ ‖b‖ sin(θ) n

2. Right-hand Rule

We can also find the direction of a cross-product vector using the right-hand rule. Place your index finger, index finger and thumb so that they all form up like an X, Y, and Z coordinate system. Then move your hand to the point that your index finger points to vector a and your middle finger points to vector b. Your thumb will indicate the direction of the cross product vector.