#### RESULTS

Having trouble figuring out the sum of two trajectories? No worries, you’re at the right place! Our free normal vector calculator is here to save the day for you. Simply input values in this tool and you’re good to go!

## What is Cross-Product

To understand it, let’s first get our heads around vector: It is a mathematical tool with a clearly defined magnitude and direction. It is utilized in Physics, Mathematics, Engineering and Computer Science.

The cross-product (don’t confuse with dot-product), to put it simply, is a binary operation on two trajectories in 3-dimensional space. It is represented by the sign ‘x’ (read: Cross). Consider two linearly self-determining of these, ‘a’ and ‘b’; the cross product of these two vectors would be a trajectory that is perpendicular to both a and b.

## Cross-Product Formula

Again, let’s consider the ‘a x b’ with ‘x’ being the result of their multiplication.

So, the formula goes like this:

`\(\mathbf{C = a \times b = |a| |b| sin(\theta) n}\)`

Where,

`\(C=?\)`

**a** – is the first vector,

**b** – is the second trajectory,

**θ** - is an angle between both the above-mentioned vectors,

**n** – the resulting third path-line perpendicular to both a and b.

## How to do Cross-Product

If you know how to work out the multiplications, then it becomes quite simple actually. All you need to do is use the above-stated formula and you’re good to go. Additionally, as you may already know by now that the result of two path-lines is a third route that is at a right angle to both the previous vectors.

However, it is pertinent to note here that when both the trajectories ‘a’ and ‘b’ point in the same or the opposite direction, the length of the third path-line is 0. However, when both ‘a’ and ‘b’ are at right angle relative to each other, the length of the third one is maximized.

Let’s define ‘a’ and ‘b’ with coordinates ax, ay and az and bx, by and bz respectively. Now naturally, let’s assume that the resultant vector ‘c’ goes with coordinates cx, cy and cz.

Consider the values:

`\(a = (4,5,6)\)`

`\(b = (7,8,9)\)`

`\(c =?\)`

Let’s do the math and find out!

`\(\mathbf{cx = aybz - azby} = 5\times9 - 6\times8 = 45 - 48 = -3\)`

`\(\mathbf{cy = azbx - axbz} = 6*7 - 4*9 = 42 - 36 = 6\)`

`\(\mathbf{cz = axby - aybx} = 4*8 - 5*7 = 32 - 35 = -3\)`

your answer = `\(-3,6,-3\)`

Or, if you aren’t interested in doing all the math, you can simply use our cross multiply calculator and get the answer automatically in a split second.

## Cross-Product of two vectors

The two vectors ‘a’ and ‘b’ follow the rules given below:

`\((ya) x b=y (a x b)=a x (yb)\)`,

`\(a x (b+c)=a × b+a × c\)`

`\((b+c) × a=b × a+c × a\)`

Where c is the sum value while y is a scaler. We can use these properties, to come up with a formula for the multiplication result in relation to components.

## How to use Cross-Product Calculator

Coming back to our digital gizmo, you can determine the path-line results with our vector multiplication calculator. It is extremely easy to use. There are two ways you can determine the answer. Either you can use, coordinates method or the initial points method.

Both options are given in our cartesian product calculator. You can select either of these two options for the purpose of calculation.

## Coordinates Method and Initial points Method

- In Coordinates approach, you have to input the coordinates (x,y,z) for the unit trajectories whose product trajectory you want to determine and that is that.
- On the other hand, in initial points approach, you have to input the initial points as well as the terminal points of both the unit path-lines to get the answer.

All that you have to do is simply follow the steps given below:

- Input the values (Coordinates or initial points) of the two trajectories
- Click on ‘calculate’ to find the answer.

## dot product vs cross product

People often wonder *“Is dot product same as the cross product?”* these two are polar opposites. The dot multiplication is scaler in nature and a scaler is not defined by a particular direction while vector on the other hand, is described with a specific direction.