Q.Find the vector and the Cartesian equations of the line through the point $(5, 2, -4)$ and which is parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$.

Concept: Vector Equation of a Line — a line through a fixed point with position vector $\vec{a}$ and parallel to a direction vector $\vec{b}$ is given by $\vec{r} = \vec{a} + \lambda \vec{b}$.

Step 1: Identify the given point and direction vector.

Point: $(5, 2, -4)$ gives $\vec{a} = 5\hat{i} + 2\hat{j} - 4\hat{k}$.

Direction: $\vec{b} = 3\hat{i} + 2\hat{j} - 8\hat{k}$.

Step 2: Write the vector equation directly:

Step 3: For the Cartesian equation, equate components: …

Concept and Intuition

A line in space is completely determined once we know two things: a point it passes through, and its direction. The vector equation of a line is the most natural way to express this — it says "start at the given point, then move any distance along the direction vector." Every point on the line corresponds to some scalar multiple of the direction vector added to the position vector of the fixed point.

The Cartesian equations (also called symmetric equations) simply unpack this vector idea into separate coordinates. They tell you: for any point on the line, the displacement from the fixed point in each coordinate direction is proportional to the corresponding component of the direction vector.

Step-by-step solution

1. Write the position vector of the given point

The point is $(5, 2, -4)$. Its position vector is:

2. Identify the direction vector

The line is parallel to $3\hat{i} + 2\hat{j} - 8\hat{k}$, so this is our direction vector:

3. Write the vector equation

The vector equation of a line through point $\vec{a}$ parallel to $\vec{b}$ is:

Substituting:

This is the vector equation. The parameter $\lambda$ can be any real number — each value gives a distinct point on the line.

4. Convert to Cartesian (symmetric) equations

If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then equating components from the vector equation:

$x = 5 + 3\lambda, \quad y = 2 + 2\lambda, \quad z = -4 - 8\lambda$ …

Method: Vector and Cartesian Equation of a Line Through a Point, Parallel to a Vector

Use this when a line is specified by one point it passes through and a vector it is parallel to, and you need both the vector and Cartesian (symmetric) forms.

Steps

Step 1: Identify $\vec{a}$ and $\vec{b}$.

The given point gives the position vector $\vec{a}$; the "parallel to" vector is the direction $\vec{b}$.

Step 2: Write the vector equation.

Step 3: Unpack into Cartesian form. …

Common Mistakes

Mistake 1: Dropping the negative sign in the $z$-denominator of the Cartesian form.

Why it's wrong: the direction's $z$-component is $-8$, so the denominator is $-8$; writing $\frac{z+4}{8}$ describes a different line. Correct approach: keep $\frac{z+4}{-8}$.

Mistake 2: Getting the numerator sign wrong for a negative point coordinate. …