Question

The cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$. Write its vector form.

Answer 1

The Cartesian equation of the line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$……….....(1)
The given line passes through the point (5,-4,6).
The position vector of this point is $\vec a = 5\hat i-4\hat j +6 \hat k$
Also, the direction ratios of the given line are 3, 7, and 2.

This means that the line is in the direction of the vector, $\vec b$=3$\hat i$+7$\hat j$+2$\hat k$

It is known that the line through position vector $\vec a$ and in the direction of the vector $\vec b$ is given by the equation,
$\vec r$=$\vec a$+λ$\vec b$, λ∈R
⇒ $\vec r$=(5$\vec i$-4$\vec j$+6$\vec k$)+λ(3$\vec i$+7$\vec j$+2$\vec k$)

This is the required equation of the given line in vector form.