Key Concepts and Formulas

• Vector Representation of a Line Segment: A line segment joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ can be represented by a vector $\overrightarrow{PQ} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$.
• Scalar Projection of a Vector: The scalar projection of a vector $\vec{a}$ onto another vector $\vec{b}$ is a scalar quantity representing the component of $\vec{a}$ along the direction of $\vec{b}$. It is given by the formula: $\text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ Here, $\vec{a} \cdot \vec{b}$ is the dot product of vectors $\vec{a}$ and $\vec{b}$, and $|\vec{b}|$ is the magnitude of vector $\vec{b}$.
• Dot Product: For $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ and $\vec{b} = b_x\hat{i} + b_y\hat{j} + b_z\hat{k}$, their dot product is $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$.
• Magnitude of a Vector: The magnitude of a vector $\vec{b} = b_x\hat{i} + b_y\hat{j} + b_z\hat{k}$ is given by $|\vec{b}| = \sqrt{b_x^2 + b_y^2 + b_z^2}$.

Step-by-Step Solution

We need to find the projection of the line segment joining points A and B onto the line joining points C and D. This means we will find the scalar projection of vector $\overrightarrow{AB}$ onto vector $\overrightarrow{CD}$.

Step 1: Define the Points and Form the Vectors

Let the given points be:

• $A = (1, -1, 3)$
• $B = (2, -4, 11)$
• $C = (-1, 2, 3)$
• $D = (3, -2, 10)$

1.1. Calculate Vector $\overrightarrow{AB}$ (the vector to be projected): This vector represents the line segment from A to B. $\overrightarrow{AB} = (2 - 1)\hat{i} + (-4 - (-1))\hat{j} + (11 - 3)\hat{k}$ $\overrightarrow{AB} = (1)\hat{i} + (-4 + 1)\hat{j} + (8)\hat{k}$ $\overrightarrow{AB} = \hat{i} - 3\hat{j} + 8\hat{k}$

1.2. Calculate Vector $\overrightarrow{CD}$ (the vector representing the line onto which projection is made): This vector represents the direction of the line from C to D. $\overrightarrow{CD} = (3 - (-1))\hat{i} + (-2 - 2)\hat{j} + (10 - 3)\hat{k}$ $\overrightarrow{CD} = (3 + 1)\hat{i} + (-4)\hat{j} + (7)\hat{k}$ $\overrightarrow{CD} = 4\hat{i} - 4\hat{j} + 7\hat{k}$

Step 2: Calculate the Dot Product of the Vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$

We use the dot product formula $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$. For $\overrightarrow{AB} = \hat{i} - 3\hat{j} + 8\hat{k}$ and $\overrightarrow{CD} = 4\hat{i} - 4\hat{j} + 7\hat{k}$: $\overrightarrow{AB} \cdot \overrightarrow{CD} = (1)(4) + (-3)(-4) + (8)(7)$ $\overrightarrow{AB} \cdot \overrightarrow{CD} = 4 + 12 + 56$ $\overrightarrow{AB} \cdot \overrightarrow{CD} = 72$

Step 3: Calculate the Magnitude of the Vector $\overrightarrow{CD}$

We need the magnitude of the vector onto which the projection is made, which is $\overrightarrow{CD}$. Using the magnitude formula $|\vec{b}| = \sqrt{b_x^2 + b_y^2 + b_z^2}$ for $\overrightarrow{CD} = 4\hat{i} - 4\hat{j} + 7\hat{k}$: $|\overrightarrow{CD}| = \sqrt{(4)^2 + (-4)^2 + (7)^2}$ $|\overrightarrow{CD}| = \sqrt{16 + 16 + 49}$ $|\overrightarrow{CD}| = \sqrt{81}$ $|\overrightarrow{CD}| = 9$

Step 4: Apply the Projection Formula

Now, substitute the calculated dot product and magnitude into the scalar projection formula: $\text{Projection of } \overrightarrow{AB} \text{ on } \overrightarrow{CD} = \frac{\overrightarrow{AB} \cdot \overrightarrow{CD}}{|\overrightarrow{CD}|}$ $\text{Projection} = \frac{72}{9}$ $\text{Projection} = 8$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with negative signs, especially when subtracting coordinates or squaring negative numbers. For example, $(-4)^2 = 16$, not $-16$.
• Order of Subtraction: When forming a vector $\overrightarrow{PQ}$ from points $P$ and $Q$, always subtract the coordinates of the initial point $P$ from the final point $Q$ (i.e., $Q-P$).
• Distinguish Scalar vs. Vector Projection: The problem asks for "the projection," which typically implies the scalar projection. If vector projection were asked, the formula would be $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$.

Summary

To find the projection of one line segment onto another, we first form vectors representing these segments. Then, we calculate the dot product of the vector being projected and the magnitude of the vector onto which it is projected. Finally, we divide the dot product by the magnitude to get the scalar projection. Following these steps, the projection of the given line segment is 8.

The final answer is \boxed{8}.