Animated Solution for Mathematics - Three Dimensional Geometry: The length of the projection of the line segment joining the points $(5,-1,4)$ and $(4,-1,3)$ on the plane, $x+y+z=7$ is :

The problem asks for the length of the projection of a line segment $AB$ onto the plane defined by $x+y+z=7$. We are given the points $A(5,-1,4)$ and $B(4,-1,3)$.

First, we determine the vector $\overrightarrow{AB}$ by subtracting the coordinates of $A$ from $B$:

The magnitude of this vector, representing the true length of the segment in space, is:

Every plane is defined by its normal vector $\vec{n}$. For the plane $x+y+z=7$, the coefficients of $x,y,$ and $z$ provide the normal vector:

The magnitude of this normal vector is:

To find the length of the projection $L$, we construct a right-angled triangle. The hypotenuse is the original segment $AB$, the base is the projection $L$, and the height $h$ is the perpendicular distance between the two parallel lines formed by the projections of $A$ and $B$.

The height $h$ is the scalar projection of $\overrightarrow{AB}$ onto the normal vector $\vec{n}$, calculated as:

We compute the dot product:

Taking the absolute value and dividing by the magnitude of the normal vector, we find:

Applying the Pythagorean theorem, $L^2+h^2=|\overrightarrow{AB}{|}^2$, we substitute our known values:

This simplifies to:

Thus, the length of the shadow is:

$$L=\sqrt{\frac{2}{3}}$$