Question

Find the vector and Cartesian equations of a line which passes through the point \((1,2,3)\) and is parallel to the vector \(2\hat i+3\hat j+2\hat k\).

Hint:

Use r = a + lambda b for the vector form; equate components for the Cartesian symmetric form.

Answer 1

Step 1: Identify the point vector and direction vector:
Point $(1,2,3)$ gives $\vec a=\hat i+2\hat j+3\hat k$; the given parallel vector is the direction $\vec b=2\hat i+3\hat j+2\hat k$.

Step 2: Parametrize each coordinate separately:
$x=1+2\lambda$, $y=2+3\lambda$, $z=3+2\lambda$ — this already is the vector equation written coordinate-wise.

Step 3: Eliminate the parameter $\lambda$:
Solving each for $\lambda$: $\lambda=\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{2}$.

Final Answer:
\[ \boxed{\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+2\hat k);\quad \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{2}} \]