Q6:

Find the cartesian equation of the line which passes through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.

Step 1: Identification of Given Parameters

The problem requires the determination of the Cartesian equation of a line passing through a fixed point $A(x_1, y_1, z_1)$ and parallel to a given vector $\vec{b}$.

• Point $A$ through which the line passes: $(-2, 4, -5)$. Thus, $x_1 = -2$, $y_1 = 4$, $z_1 = -5$.
• The given line equation is $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.
• The direction ratios of the given line are $(a, b, c) = (3, 5, 6)$. [Per the standard form of a line equation $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$].

Step 2: Establishing the Parallel Condition

Since the required line is parallel to the given line, their direction ratios must be proportional. In this specific case, the required line can share the exact same direction ratios as the given line, $\vec{v} = 3\hat{i} + 5\hat{j} + 6\hat{k}$.

[Justification: Two lines are parallel if and only if their direction vectors are scalar multiples of each other. By choosing the same direction vector $(3, 5, 6)$, we satisfy the condition of parallelism.]

Step 3: Formulation of the Cartesian Equation

The Cartesian equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ is given by the formula:

$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$

Substituting the identified values:

• $x_1 = -2 \implies x - (-2) = x + 2$
• $y_1 = 4 \implies y - 4$
• $z_1 = -5 \implies z - (-5) = z + 5$
• $a = 3, b = 5, c = 6$

Step 4: Geometrical Representation

Step 5: Final Equation Assembly

Combining the variables into the standard Cartesian form:

$$\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}$$

Final Solution: The Cartesian equation of the line is $\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}$.