Q.The vector equation of the line through the points $(3, 4, -7)$ and $(1, -1, 6)$ is __________.

Concept: Vector Equation of a Line — a line through two points $A$ and $B$ is given by $\vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a})$, where $\vec{a}$ and $\vec{b}$ are position vectors of the points.

Step 1: Let $\vec{a} = 3\hat{i} + 4\hat{j} - 7\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + 6\hat{k}$.

Step 2: Direction vector $\vec{d} = \vec{b} - \vec{a} = (1-3)\hat{i} + (-1-4)\hat{j} + (6+7)\hat{k} = -2\hat{i} -5\hat{j} + 13\hat{k}$.

Step 3: Using $\vec{r} = \vec{a} + \lambda \vec{d}$, the equation is: …

To write the vector equation of a line, you need two things: a position vector of any point on the line, and a direction vector that tells you which way the line runs. Think of it like this: you start at a fixed point (the position vector), and then you can move any distance along the direction vector — that sweep gives you every point on the line.

The general form is:

Now let’s apply this to the given points.

1. Pick a point on the line. You can choose either of the two given points. Let’s take $(3, 4, -7)$. Its position vector is:

(If you chose the other point, the final equation would look different but describe the same line — that’s fine.)

2. Find the direction vector. The direction vector is simply the vector from one point to the other. Subtract the coordinates of the first point from the second (or vice versa — the direction just flips sign, which is still parallel). From $(3,4,-7)$ to $(1,-1,6)$:

So $\vec{b} = (-2, -5, 13)$. …

Method: Vector Equation of a Line Through Two Given Points

Use this whenever a line is fixed by two points it passes through rather than by a stated direction.

Steps

Step 1: Turn each point into a position vector.

For points $A$ and $B$, write $\vec{a}$ and $\vec{b}$ as $x\hat{i}+y\hat{j}+z\hat{k}$.

Step 2: Build the direction from the two points.

A line's direction is the displacement from one point to the other:

Subtract componentwise. Reversing the order only flips the sign, and $-\vec{d}$ runs the same line the opposite way, so either order is valid. Watch double negatives, e.g. $6-(-7)=13$.

Step 3: Write the equation with one point plus the direction. …

Common Mistakes

Mistake 1: Using a point's position vector as the direction.

Why it's wrong: a single point gives no direction; the line's direction comes from the difference of the two points, $\vec{b}-\vec{a}$. Correct approach: always subtract the two position vectors to get the direction before writing the equation.

Mistake 2: Subtracting the coordinates inconsistently. …