Solving Scalar Product Problem: Find Perpendicular Distance

[pavadrin]

Hey,
Today I was given a problem to solve in class and was told to complete it for homework. This problem is as follows:

The line y=mx + c has a gradient m and cuts the y-axis at (0,c). Thus we can write the parametric vector equation of the line as:

$$r = cj +\lambda (i + mj)$$

Using this fact show that that the perpendicular distance from point $$A(x_1 , y_1)$$ to y = mx + c is:

$$\mid(\frac{mx_{1} - y_{1} + c}{\sqrt{m^2 + 1}})\mid$$

If y = mx + c is instead written as ax + by + d = 0 show that the perpendicular distance of point $$A(x_1 , y_1)$$ to as ax + by + d = 0 is given by:

$$\mid(\frac{ax_{1} - by_{1} + d}{\sqrt{a^2 + b^2}})\mid$$

This diagram which I drew to help me may help:

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I have tried solving this problem by using vectors:

and I know that the dot product of $$( x_1 , y_1 )$$ and y = mx + c is equal to zero but from there onwards I am not sure on how to approach this problem. All help is appreciated,
thanks, Pavadrin

 

[e(ho0n3]

First, you need to find a vector perpendicular to the line y = mx + c. Call this vector u. Let v be the vector that points to A. Then for some n, nu + v = r. The magnitude of nu is the perpendicular distance from the point A to the line y = mx + c.

 

[Architeuthis Dux]

Hello Pavadrin,

Solving scalar product problems can be challenging, but with the right approach, it can be solved easily. The first step is to understand the problem and the given equations. In this case, we are looking for the perpendicular distance from a point to a given line.

To solve this problem, we can use the formula for the perpendicular distance from a point to a line. This formula is derived from the dot product of two vectors. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. In this case, the angle between the two vectors is 90 degrees, so the cosine is equal to 0. Therefore, the dot product is equal to 0, which gives us the formula for the perpendicular distance.

Using this formula, we can solve for the perpendicular distance from point A to y = mx + c. We substitute the values of point A and the given line equation into the formula and simplify it to get the final result:

Perpendicular distance = |(mx_1 - y_1 + c)/√(m^2 + 1)|

Similarly, for the line ax + by + d = 0, we can use the same formula and substitute the values to get the final result:

Perpendicular distance = |(ax_1 - by_1 + d)/√(a^2 + b^2)|

I hope this helps you in solving the problem. Remember, understanding the concept and using the right formula is key in solving any scalar product problem. Keep practicing and you will master it in no time.

Best regards,