The projection of the line segment joining the points (1, -1, 3) and (2, -4, 11) on the line joining

Q: Wait, does the position of the line $CD$ matter, or just its direction?

Great question! In vector projection, we only care about the direction of the line we are projecting onto. The line $CD$ acts as a vector $\vec{CD}$. The specific coordinates of $C$ and $D$ are only used to find the direction vector $\vec{CD} = \vec{D} - \vec{C}$. The actual position of the line in space doesn't change the length of the shadow cast upon it.

Q: Why do we use the absolute value in the formula $\frac{|\vec{a} \cdot \vec{b}|}{|\vec{b}|}$?

Because we are looking for a length. Length must always be a non-negative scalar. The dot product $\vec{a} \cdot \vec{b}$ can be negative if the angle between the vectors is obtuse (greater than $90^\circ$). By taking the absolute value, we ensure the projection length is always positive, regardless of the orientation of the vectors.

Q: Can I use the projection of $A$ and $B$ separately onto the line?

You absolutely can! You could find the projection of point $A$ onto the line $CD$ (let's call it $A'$) and the projection of point $B$ onto the line $CD$ (let's call it $B'$). Then, the distance between $A'$ and $B'$ would be the length of the projection. However, that is a much longer, more tedious process involving finding the equation of the line and the foot of the perpendicular. The vector method we used is the elegant, 'JEE-smart' way to get the same result in seconds.

Question

The projection of the line segment joining the points $(1, -1, 3)$ and $(2, -4, 11)$ on the line joining the points $(-1, 2, 3)$ and $(3, -2, 10)$ is:

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast, three-dimensional space. You have a line segment $AB$ suspended in the air, and you need to understand how it 'looks' from the perspective of another line $CD$. This is the essence of projection.

In the world of JEE Advanced, this isn't just about drawing lines; it's about understanding the relationship between vectors. We start by defining our segment $AB$ as a vector.

Given points $A(1, -1, 3)$ and $B(2, -4, 11)$, we calculate the vector $\vec{AB}$ by subtracting the coordinates of $A$ from $B$:

$$\vec{AB} = (2-1)\hat{i} + (-4 - (-1))\hat{j} + (11-3)\hat{k} = \hat{i} - 3\hat{j} + 8\hat{k}$$

This vector represents the displacement from $A$ to $B$. Now, we turn our attention to the target line $CD$, defined by points $C(-1, 2, 3)$ and $D(3, -2, 10)$.

Just as we did for $AB$, we find the direction vector $\vec{CD}$:

$$\vec{CD} = (3 - (-1))\hat{i} + (-2-2)\hat{j} + (10-3)\hat{k} = 4\hat{i} - 4\hat{j} + 7\hat{k}$$

### The Physics of the Shadow

Think of the projection as a shadow. If you shine a light perpendicular to the line $CD$, the segment $AB$ casts a shadow onto $CD$. The length of this shadow is what we are after.

Mathematically, this is the scalar projection of $\vec{AB}$ onto $\vec{CD}$. The formula is elegant and powerful:

$$P = \frac{|\vec{AB} \cdot \vec{CD}|}{|\vec{CD}|}$$

Why this formula? The dot product $\vec{AB} \cdot \vec{CD}$ measures how much $\vec{AB}$ aligns with $\vec{CD}$, and dividing by the magnitude $|\vec{CD}|$ normalizes the direction, giving us the length of the component of $\vec{AB}$ along $\vec{CD}$.

### The Calculation

Let's execute this. First, we calculate the dot product:

$$\vec{AB} \cdot \vec{CD} = (1)(4) + (-3)(-4) + (8)(7) = 4 + 12 + 56 = 72$$

Next, we find the magnitude of the target vector $\vec{CD}$:

$$|\vec{CD}| = \sqrt{4^2 + (-4)^2 + 7^2} = \sqrt{16 + 16 + 49} = \sqrt{81} = 9$$

Finally, we bring it all together to find the projection length:

$$P = \frac{72}{9} = 8$$

The beauty of this approach is its efficiency. Instead of dealing with complex coordinate geometry, we reduced the problem to a simple dot product and a magnitude calculation.

This is the kind of mathematical elegance that JEE examiners love. Remember, whenever you see a projection problem, don't panic. Visualize the shadow, define your vectors, and let the dot product do the heavy lifting for you. The final answer is **8**.

Animated Solution for Mathematics - Vector Algebra: The projection of the line segment joining the points $(1, - 1, 3)$ and $(2, - 4, 11)$ on the line joining the points $(- 1, 2, 3)$ and $(3, - 2, 10)$ is:

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Visualized Solution

Identify the Segment $A B$

Calculate Vector $A B$

Identify the Target Line $C D$

Calculate Vector $C D$

The Concept of Projection

The Projection Segment

The Projection Formula

Calculate the Dot Product

Calculate the Magnitude of $C D$

Final Projection Calculation

The Sigma Insight: Scalar (Dot) Product

Analyzing the Setup

The Physics of the Shadow

The Calculation

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