Q.State whether the following statement is True or False: The vector equation of the line $\dfrac{x-5}{3} = \dfrac{y+4}{7} = \dfrac{z-6}{2}$ is $\vec{r} = 5\hat{i} - 4\hat{j} + 6\hat{k} + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k})$.

Concept: Vector Equation Of Line — converting symmetric form to vector form.

Step 1: The symmetric form $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ gives a point on the line: $(5, -4, 6)$, i.e. $5\hat{i} -4\hat{j} +6\hat{k}$.

Step 2: The direction ratios are $(3, 7, 2)$, so the direction vector is $3\hat{i} +7\hat{j} +2\hat{k}$. …

The core idea here is the translation between Cartesian and vector forms of a line. Every line in 3D can be written as:

where $\vec{a}$ is the position vector of a fixed point on the line, and $\vec{b}$ is a vector parallel to the line (the direction vector). The Cartesian form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ is just another way of saying the same thing — the denominators are the direction ratios, and $(x_1, y_1, z_1)$ is a point on the line.

So the question is simply: does the given vector equation correctly extract these two pieces from the Cartesian equation?

1. Identify the fixed point from the Cartesian form.

   The given line is $\dfrac{x-5}{3} = \dfrac{y+4}{7} = \dfrac{z-6}{2}$.

   In the standard form $\dfrac{x-x_1}{a}$, the numerator is $(x - x_1)$. Here $x-5$ means $x_1 = 5$.

   For $y$, we have $y+4$ which is $y - (-4)$, so $y_1 = -4$.

   For $z$, $z-6$ gives $z_1 = 6$.

   So the point is $(5, -4, 6)$, whose position vector is $5\hat{i} - 4\hat{j} + 6\hat{k}$.

   This matches the $\vec{a}$ in the given vector equation.

2. Identify the direction vector.

   The denominators are $3, 7, 2$ — these are the direction ratios.

   So a direction vector parallel to the line is $3\hat{i} + 7\hat{j} + 2\hat{k}$. …

Method: Checking a Symmetric-to-Vector Line Conversion (True/False)

When a statement claims a given vector equation matches a symmetric-form line, don't re-derive blindly — verify the two ingredients a line needs.

Steps

Step 1: Extract the true point and direction from the symmetric form.

Rewrite each fraction as $\dfrac{x-x_1}{a}$ so signs are unambiguous (a term $y+4$ means $y_1=-4$). The numerator constants give the point $(x_1,y_1,z_1)$; the denominators give the direction ratios $(a,b,c)$.

Step 2: Read the point and direction the statement offers.

In $\vec{r}=\vec{a}+\lambda\vec{b}$, the constant vector is the point and the $\lambda$-coefficient is the direction. …

Common Mistakes

Mistake 1: Misreading the sign of the $y$-coordinate.

Why it's wrong: the symmetric form has $y+4=y-(-4)$, so the point's $y$-coordinate is $-4$; reading it as $+4$ would wrongly call a correct statement false. Correct approach: rewrite each numerator as $x-x_1$ before judging the statement.

Mistake 2: Rejecting a valid answer because the direction was scaled. …