Calc 3: find equation in space?

[whig4life]

Suppose a, b are two positive numbers. Find an equation for the line in space
passing through "a" on the x-axis and "b" on the y-axis.

Show steps and explain how you arrived at this.

 

[tkhunny]

Do you know the "Intercept Form"?

$\displaystyle \frac{x}{a} + \frac{y}{b} = 1$

Done.

 

[HallsofIvy]

Suppose a, b are two positive numbers. Find an equation for the line in space
passing through "a" on the x-axis and "b" on the y-axis.

Show steps and explain how you arrived at this.

A line can be written in a variety of ways: y= mx+ b, Ax+ By= 1, x= py+ q, etc. Each of those involves two coefficients that you need to find. You need two equations to find two values.

Every point on the x-axis has y value 0 and every point on the y- axis has x value 0 so saying the line passes "through "a" on the x-axis and "b" on the y-axis" means that x= a, y= 0 and x= 0, y= b satisfy the equation. Choose any one of the forms above, set x= a, y= 0 in it to get one equation, set x= 0 y= b in it to get another equation, and solve for the coefficients.

 

[soroban]

Hello, whig4life!

Did I interpret the question correctly?

Suppose $\displaystyle a, b$ are two positive numbers.
Find an equation for the line in space
passing through $\displaystyle a$ on the x-axis and $\displaystyle b$ on the y-axis.

Show steps and explain how you arrived at this.

The x-intercept is $\displaystyle A(a,0,0)$
The y-intercept is $\displaystyle B(0,b,0)$

The vector $\displaystyle \vec v \:=\:\overrightarrow{AB} \:=\:\langle \text{-}a,b,0\rangle$


The line through $\displaystyle (a,0,0)$ with direction vector $\displaystyle \vec v \:=\:\langle \text{-}a,b,0\rangle$

. . . has the parametric equations: .$\displaystyle \begin{Bmatrix}x \:=\: a-at \\ y \:=\: bt \\ z \:=\: 0 \end{Bmatrix}$

 

[mmm4444bot]

Hi whig.

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