When you have the correct tools, you can find the coordinates with distances and angles. But where do you start? To find the vector, you have to know where to begin the measuring in order to solve the problem. When you have the distance between a particular point and its mate, or at least another point to pinpoint for your problem, then you have a magnitude. The reference angle to that point gives you the correct direction. Combining the magnitude with the direction gives you a correct vector.

### Second Points

Figuring out the coordinates, math problem follows, of the second point requires vector analysis. To begin to get to the correct answer, gather the known quantities of the problem. These will include the reference coordinates, angle and distance. Then you calculate the cosine of the angle. Once that is figured, multiply that by the distance. The sine of the angle is calculated. Multiply this by the distance. Add the cosine result to the x-coordinate. This is done from the starting point. Add the sine result to the y-coordinate in order to conclude the coordinates for the second point.

An example would be using Point A on the coordinate 4 and Point B on coordinate -1 on a number line.

_*A_B*=|4−(−1)|=|4+1|=|5|=5

To find the distance between two points in a coordinate plane, a different formula based on the Pythagorean Theorem is used. This is seen as (x_{1},y_{1}) and (x_{2},y_{2}) are the coordinates. The distance is marked as d.

* D= /(x*2−

*1)2+(*

*x**2−*

*y**1)2*

*y*### What Is a Vector?

A vector has size, also known as magnitude, and direction. Scalars, physical and geometric quantities, are typically defined by stipulating their magnitude in appropriate units of measure. Graphically, scalars can be represented by points on a numerical scale. For instance, mass is expressed in grams and temperature in degrees. The basic distance formula is used in order to find the distance on a number line between two points. That formula is: AB=|b−a|or|a−b|.

### Pythagorean Theorem

Named after the ancient Greek mathematician Pythagoras, you can’t complete how to find the coordinates with distances and angles without the Pythagorean theorem, also known as Pythagoras’ theorem:

a^{2} + b^{2} = c^{2}

This long held theory states that the square of the hypotenuse, which is the side that is opposite of the right angle of a triangle, is equal to the sum of the squares of the other two sides of the triangle. This is written out as an equation, called the Pythagorean equation, using a, b and c. The c represents the length of the hypotenuse. The a and b sides represent the lengths of the triangle’s other two sides.

### How to Check Your Calculations

You aren’t done just yet. No good math problem is done unless it has been proven, to a point. The Pythagorean theorem lets you check your answer. The square cosine result added to the square of the sine result should equal the square of the distance value. Make sure to include a given round-off error.

As an example, two sides of a hypotenuse of a right triangle are 4 inches and 5 inches. To find the length of the final side you would use the Pythagorean theorem as seen below:

^{}

*c*^{2} = 4^{2} + 5^{2}

*c*^{2} = 16 + 25

*c*^{2} = 41

*c* = /41

c = 6.40312423743 inches

If you have a calculator, enter the length of the known sides and follow the Pythagorean theorem to get your answer.

#### Things You Will Need

- Pen or pencil
- Scientific calculator

#### Tips

- You can check your calculations using the Pythagorean theorem. The square cosine result plus the square of the sine result should equal the square of the distance value, within a given round-off error.
- (21.563)^2 + (16.249)^2 = (27)^2
- 464.963 + 264.030 = 728.993
- 728.993 = 729