1. ## equation

A bird flies linearly, according to the equation
(x,y,z) = (5,6,7) + t[2,3,1]. Assume that the Sun is directly overhead , making the Sun's rays perpendicular to the xy-plane which represents the ground. The birds shadow is said to be projected perpendicularly onto the level ground. Find an equation that describes the motion of the shadow.

I got y=3/2x- 3/2

2. Hello, Veronica1999!

A bird flies linearly, according to the equation: . $(x,y,z) \:=\: (5,6,7) + t[2,3,1]$
Assume that the Sun is directly overhead, making the Sun's rays perpendicular
to the xy-plane, which represents the ground.
The bird's shadow is said to be projected perpendicularly onto the level ground.
Find an equation that describes the motion of the shadow.

I got: . $y\:=\:\tfrac{3}{2}x - \tfrac{3}{2}$ . Correct!

We have: . $\begin{Bmatrix}x &=& 5 + 2t & [1] \\ y &=& 6+3t & [2] \\ z &=& 0 & [3] \end{Bmatrix}$

From [1]: . $x \:=\:5+2t \quad\Rightarrow\quad t \:=\:\frac{x-5}{2}\;\;[4]$

From [2]: . $y \:=\:6+3t \quad\Rightarrow\quad t \:=\:\frac{y-6}{3}\;\;[5]$

Equate [4] and [5]: . $\frac{x-5}{2} \:=\:\frac{y-6}{3} \quad\Rightarrow\quad y \:=\:\tfrac{3}{2}x - \tfrac{3}{2}$

3. I did this in a slightly different way (only slightly):
From x= 5+ 2t, t= (x- 5)/2.

Then y= 6+ 3t= 6+ 3(x- 5)/2= 6+ (3/2)x- 15/2= 12/2+ (3/2)x- 15/2= (3/2)x- 3/2

Saying that the projection is perpendicular means that we can just ignore z.