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: Write the vector equation:
A line through point \(\vec a=\hat i+2\hat j+3\hat k\) parallel to \(\vec b=2\hat i+3\hat j+2\hat k\) has vector equation \(\vec r=\vec a+\lambda\vec b\).
\[ \vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+2\hat k) \]

Step 2: Convert to Cartesian form:
With \(\vec r=x\hat i+y\hat j+z\hat k\), matching components gives \(x=1+2\lambda,\ y=2+3\lambda,\ z=3+2\lambda\), so
\[ \dfrac{x-1}2=\dfrac{y-2}3=\dfrac{z-3}2 \]

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