Animated Solution for Mathematics - Three Dimensional Geometry: If $(a,b,c)$ is the image of the point $(1,2,-3)$ in the line $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}$, then $a+b+c$ is equal to:

A Journey of Reflection

Imagine you are standing in a room, holding a point $P(1,2,-3)$ in your hand. Before you lies a mirror, but this mirror is not a flat surface—it is a straight line $L$ defined by the equation:

Your goal is to find the reflection of point $P$ on the other side of this line. This is not just a calculation; it is a beautiful dance of vectors and symmetry.

The Stepping Stone

To find the image $R(a,b,c)$, we cannot simply jump across the line. We need a bridge. That bridge is the foot of the perpendicular, $Q$.

If we drop a perpendicular from $P$ onto the line $L$, the point where it hits, $Q$, is the exact midpoint between $P$ and its image $R$. Think of $Q$ as the anchor point that holds the symmetry together.

The Parametric Power

Since $Q$ lies on the line $L$, we can describe its position using a single parameter, $\lambda$. By setting the line equation equal to $\lambda$, we can express any point on the line as:

This is the power of parametric form—it turns a 3D line into a simple 1D path.

The Perpendicularity Condition

Now, we construct the vector $\overrightarrow{PQ}$ by subtracting the coordinates of $P$ from $Q$:

We know that $\overrightarrow{PQ}$ must be perpendicular to the line $L$. The direction vector of $L$ is $\vec{d}=(2,-2,-1)$.

Because they are perpendicular, their dot product must be zero: $\overrightarrow{PQ}\cdot \vec{d}=0$. This gives us the equation:

Solving this, we find $4\lambda -4+4\lambda -2+\lambda -3=0$, which simplifies to $9\lambda -9=0$, or $\lambda =1$.

The Final Symmetry

With $\lambda =1$, we find the exact coordinates of $Q$ by substituting $\lambda$ back into our general point:

Now, we use the midpoint formula: $\frac{P+R}{2}=Q$. This implies:

Solving these gives $a=1,b=0,c=1$. The image point is $R(1,0,1)$.

Finally, the sum $a+b+c=1+0+1=2$. You have successfully navigated the mirror of 3D space!