Question

Consider the following statements
Statement I The vector equation of a line passing through two points whose position vectors are $a$ and $b$, is $r=a+λ(b−a)∀λ∈R$.
Statement II The cartesian equation of a line passing through two points $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ is $\frac{x−x_1}{x_2−x_1}=\frac{y−y_1}{y_2−y_1}=\frac{z−z_1}{z_2−z_1}$
Choose the correct option.

• A. Statement $I$ is true
• B. Statement II is true
• C. Both statements are true
• D. Both statements are false

Answer 1

Answer: (C)

I. Let $a$ and $b$ be the position vectors of two points $A\left(x_1,y_1,z_1\right)$ and $B\left(x_2,y_2,z_2\right)$, respectively that are lying on a line

Let r be the position vector of an arbitrary point $P(x,y,z)$, then $P$ is a point on the line if and only if $AP=r−a$ and $AB−b−a$ are collinear vectors. Therefore, $P$ is on the line if and only if
$r−a=λ(b−a)$
or $r=a+λ(b−a),λ∈R$...(i)
This is the vector equation of the line.
II. We have,
$r=x\hat{i}+y\hat{j}+z\hat{k}$
$a=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}$
and $b=x_2\hat{i}+y_2\hat{j}+z_2\hat{k}$
On substituting these values in Eq. (i), we get $x\hat{i}+y\hat{j}+z\hat{k}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}+λ[\left(x_2−x_1\right)\hat{i}$
$+\left(y_2−y_1\right)\hat{j}+\left(z_2−z_1\right)\hat{k}]$
Equating the like coefficients of $\hat{i},\hat{j}$ and $\hat{k}$, we get
$x=x_1+λ\left(x_2−x_1\right)$
$y=y_1+λ\left(y_2−y_1\right)$
$z=z_1+λ\left(z_2−z_1\right)$
On eliminating $λ$, we obtain
$\frac{x−x_1}{x_2−x_1}=\frac{y−y_1}{y_2−y_1}=\frac{z−z_1}{z_2−z_1}$...(ii)
which is the equation of the line in cartesian form.